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The Grand Tour

Jeremy, Richard and James embark on an adventure across the globe, driving new and exciting automobiles from manufacturers all over the world. Jeremy, Richard and James embark on an adventure across the globe, driving new and exciting automobiles from manufacturers all over the world. Jeremy, Richard and James embark on an adventure across the globe, driving new and exciting automobiles from manufacturers all over the world.

• Jeremy Clarkson
• Richard Hammond
• 398 User reviews
• 6 Critic reviews
• 2 nominations

Episodes 46

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• Trivia The opening scene cost 3.4 million dollars to produce. In total 150 cars participated, worth 26 million dollars.
• Connections Featured in Good Morning Britain: Episode dated 18 November 2016 (2016)
• Soundtracks Pumpin' Irony (uncredited) Performed by Blues Saraceno

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• Nov 18, 2016
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Jeremy Clarkson Confirms The Grand Tour’s Final Episode on Amazon Prime

TV presenter Jeremy Clarkson has confirmed the end of The Grand Tour on Amazon Prime after next year. This announcement marks the conclusion of a series beloved by fans since its inception in 2016.

Key Takeaways:

• End of The Grand Tour: Jeremy Clarkson, along with co-presenters Richard Hammond and James May , will no longer film new series of The Grand Tour after next year, following a decision by Amazon Prime’s executives. This marks the end of a popular show that started in 2016 and quickly garnered a devoted fanbase.
• Fan Reactions and Future Prospects: Fans expressed their disappointment and gratitude on social media, reflecting on the impact the show had on their lives. Despite the end of this era, there’s speculation that Amazon Prime may revive The Grand Tour with new hosts, a move reportedly welcomed by the current presenters.
• Clarkson’s Future Endeavours: Clarkson mentioned a focus on his other project, ‘Clarkson’s Farm’, and there are two more special episodes of The Grand Tour slated for release, featuring travels to Mauritania and Zimbabwe.

TV presenter Jeremy Clarkson, renowned for his role in the popular car show The Grand Tour alongside Richard Hammond and James May, has recently spoken about the future of the series. The show, which began on Amazon Prime in 2016, has been a fan favourite but will see no further series after next year. This decision, made by the streaming platform’s bosses, brings an end to a series that has been both entertaining and influential for car enthusiasts.

Clarkson shared his thoughts on Instagram, stating:

“Been a busy day. No more Grand Tour after next year but a LOT more Clarkson’s Farm. Which, this evening, is looking extremely lovely.”

This post quickly became a hub for fans to express their feelings about the series ending. The emotional reactions ranged from sadness over the show’s conclusion to appreciation for the years of entertainment it provided.

One fan commented:

“Please start a podcast with the three of you, it can just be called the news and you rant for an hour about cars.”

Others expressed their gratitude:

“The end of Clarkson, Hammond and May in whatever it’s called is a sad day. Thank you for everything you did for petrolheads. And now farmheads!”

The influence of the trio was evident in another fan’s words:

“It’s really nice coming to the comments and seeing how these 3 impacted the lives of so many. These lads got me through countless dark times with laughter and endless banter. We all knew the day was coming, and the truest of fans will wish you, Hammond, May, and indeed all of the crew that made the magic happen the absolute best in the next stage. Thank you for inspiring countless enthusiasts around the world.”

Despite Clarkson’s announcement, there’s talk that Amazon Prime might continue The Grand Tour with new hosts. An insider revealed:

“It’s a surprising decision and everyone realises it very much marks the end of an era for the three presenters.”

This potential revival aligns with the presenters’ acceptance of passing the torch to a new generation.

“The Grand Tour is one of Prime Video’s most watched shows and Jeremy, James and Richard have a devoted following. But the guys have made no bones about the fact they’re all advancing in years and they have lots of other projects to pursue,” a source informed The Sun. “They just felt like the time was right and wanted to go out on a high when the show remained popular.”

As fans prepare to bid farewell to this iconic trio, they can look forward to two more special episodes of The Grand Tour, showcasing the team’s adventures in Mauritania and Zimbabwe. This farewell follows their departure from Top Gear in 2015, after a disagreement with producers.

Alex Harrington

Jeremy clarkson's health fears: "my body doesn't work anymore" amid dementia concerns, haas f1 team embraces new fia regulations for enhanced 2024 car testing, related articles.

Clarkson’s Farm Season Two Release Date Hinted At By Source Close To Production

Jeremy Clarkson’s Love For F1 Has Skyrocketed As Max Verstappen Takes First Win

James May Reveals What Was Going Through His Head During Grand Tour Crash

Clarkson’s farm: kaleb cooper has a present for jeremy clarkson [video].

Prime Number Calculator

What is a prime number, how to use our prime number calculator, how to check if a number is prime or composite, how to find prime numbers, why do prime numbers matter, relatively prime numbers.

With our prime number calculator you can check if any given number is prime or composite. Keep reading if you want to learn what a prime number is, and how to check if a number is prime. We also discuss how to find prime numbers, as well as what relatively prime numbers are. As a bonus, we tell you if 1 is prime, and also what the oddest prime number is.

A natural number greater than 1 is called prime if it has exactly two factors, i.e., if the number is divisible only by 1 and itself.

When a natural number is greater than 1 and isn't prime, then it's called a composite number.

What about 1? 1 is neither prime nor composite as it has only one factor (itself).

Let's discuss some examples:

• 7 is prime because its only factors are 1 and 7. Indeed, none of the number preceding 7 (2, 3, 4, 5, and 6) is a factor of it, because none of these numbers divide 7 without a remainder. If you don't quite remember what that means, go to the remainder calculator .
• 8 is composite because 2 is a factor of 8, and so 8 has more factors than just 1 and 8.

Fun fact. The oddest number amongst primes is 2, as it is the only even prime: all other primes are odd!

Well, this prime number calculator is not a very complicated one. Just enter the number you wish to check into the calculator and voila! your answer will be shown below.

If your number is composite, the calculator will tell you its smallest non-trivial factor (non-trivial here means a factor greater than one).

The easiest way to verify that a given integer n is prime is to apply the so-called trial division algorithm: it consists of testing whether n is divisible by any number between 2 and n-1 . That's a lot of computations. Fortunately, the number of trials can be reduced; it is sufficient to check only the divisibility of n by prime numbers which do not exceed √n . A version of the trial division algorithm powers this prime number calculator.

Well, you cannot find every prime number, because Euclid proved sometime around 300 BC that there are an infinite number of them. If you want to find all prime numbers up to some given limit, n , you may resort to the algorithm known as the Sieve of Eratosthenes :

• Write down all numbers from 2 to n
• Start with the smallest number in our list: 2 . Circle 2 and cross out all consecutive multiples of 2 (i.e., 4, 6, 8, ..., 120 )
• Take the smallest number that is not circled or crossed out: 3 . Circle it and cross out all its further multiples: 3, 6, 9, ...
• Continue in this same way: find the smallest available number p , circle it and cross out all of its consecutive multiples
• Check if there are numbers greater than p not yet crossed out. If so, repeat Step 4. If no, we're done.

The circled numbers in the list are all the primes below n .

The main idea here is that every number assigned to p in any step of the algorithm is necessarily prime; otherwise, it would be crossed out as a multiple of some smaller, already circled, prime number.

In the animation below you see the Sieve of Eratosthenes searching for all primes up to 120 . Here's the color legend:

• red: multiples of 2
• green: multiples of 3
• blue: multiples of 5
• yellow: multiples of 7
• the remaining purple numbers are prime

As you may imagine, for REALLY BIG numbers it is quite challenging to decide whether they are prime or composite, and therefore very elaborate methods are required. The search for big primes is on-going: as of March 2020, the largest known prime number has 24,862,048 digits! Our time unit converter helps us estimate that it would take roughly 288 days (so more than three-quarters of a year) to write this number down in its entirety, provided that one digit per second is being written down!

Prime numbers are crucial in number theory because of the fundamental theorem of arithmetic : every natural number greater than 1 can be written as a product of primes in a unique way (up to the order of multiplication). In other words, primes can be thought of as the building blocks of all other natural numbers. To learn more about prime factorization, check our prime factorization calculator .

Actually, this theorem is one of the main reasons why we don't want 1 to be called a prime number. If 1 were prime, then prime factorization would no longer be unique, because we would have, e.g., 6 = 2 * 3 = 1 * 2 * 3 = 1 * 1 * 2 * 3 = ...

As for real-life applications of prime numbers, you may encounter them, e.g., in cryptography protocols, such as the well-known RSA encryption .

Do not confuse the notion of prime numbers with that of relatively prime numbers! Two natural numbers are called relatively prime (or coprime ) if there is no integer other than 1 that divides both these numbers. In other words, their Greatest Common Factor (GCF) is equal to 1.

For instance:

• 18 and 30 are not relatively prime, as they are both divisible by 3 . You may also note that their GCD is 6 .
• 18 and 35 are relatively prime: the factors of 18 read 1, 2, 3, 6, 9, 18 and that of 35 read 1, 5, 7, 35 . We see that the sole common divisor of 18 and 35 is 1 .

To learn more, visit the GCF calculator .

• Two primes are always relatively prime
• But numbers need not be prime in order to be relatively prime!

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What Is a Prime Number? How to Tell If a Number Is Prime

A prime number is a natural number that can only be divided, without a remainder, by itself and 1. In other words, a prime number has exactly two factors. For example, 13 is only divisible by 13 and 1. In contrast, a composite number is a natural number that can be divided evenly by any number besides itself and 1. A composite number has more than two factors. For example, 14 is divisible by 1, 2, 7, and 14.

Here is a list of the prime numbers up to 1000 and a look at how to tell if a number is prime.

Interesting Prime Number Facts

• The state of being prime is called primality .
• There are an infinite number of prime numbers.
• Zero and one are not prime numbers.
• Two is the only even prime number.
• Two and three are the only consecutive prime numbers.
• No prime number greater than five ends in 5.
• No prime number ends with 0.
• Goldbach Conjecture : Every even integer greater than 2 can be expressed as the sum of two prime numbers.
• Every prime number greater than 2 and 3 can be represented as 6n+1 or 6n-1.
• Prime Number Theorem : The probability that a number is prime is inversely proportional to its number of digits.
• Lemoine’s Conjecture : Any odd integer greater than 5 can be expressed as the sum of an off prime and an even semiprime. A semiprime is the product of two prime numbers.

Prime Numbers Up to 1000

The smallest prime number is 2, which is also the only even prime number. Here is a table of all the prime numbers up to 1000.

Is 1 a Prime Number?

The number 1 is not usually considered a prime number. It’s also not a composite number.

• 1 is not a prime number because it does not have exactly two positive factors.
• 1 is not a composite number because it does not have more than two factors.

Note: There are some people who argue 1 is a prime number because it’s divisible by itself and 1 (even though these two values are the same thing).

How to Tell If a Number Is Prime

There are a few different ways to tell whether or not a number is prime. The methods are called primality tests , even though some of them actually test whether a number is composite.

Basically, you test whether a number n is evenly divisible by any prime number between 2 and √ n . This is called trial division or factorization.

• No even number except 2 is prime. If a number ends with 0, 2, 4, 6, or 8, it’s a composite number.
• If the sum of the digits of a number are divisible by 3, it’s a composite number. A prime number can end with 3.
• No prime number ends with 5, except 5.
• If a number passes all of these tests, check to see if it’s divisible by prime numbers smaller than it. It’s not necessary to check prime numbers greater than √ n . Start with 3, 5, 7, 11, and work your way up to √ n .
• Check whether or not a number can be expressed as either 6n+1 or 6n-1. For example, the prime number 11 can be written as 6(2)-1.

Examples: Finding a Prime Number Using Factorization

• Is 15874 prime?
• Right away, you can see it’s not prime because it ends with an even number.
• Is 26577 a prime number?
• It does not end in 0, 2, 4, 6, 8.
• The sum of the digits 2 + 6 + 5 + 7 + 7 = 27.
• 27 is divisible by 3, so 26577 is not prime.
• Is 103 a prime number?
• It does not end in 5.
• The sum of the digits 1 + 0 + 3 = 4. It is not divisible by 3.
• The √ 103 is ~10.14. So, check to see if 103 is divisible by other primes under 10.
• 103 is not evenly divisible by 7.
• 103 is a prime number!

What Is the Largest Prime Number?

There are an infinite number of prime numbers, so computers discover new primes (slowly, because it takes a lot of computing power). To date, the largest prime number is 2 82,589,933 -1. The Great Internet Mersenne Prime Search (GIMPS) found this prime on December 7, 2018.

• Adler, Irving (1960). The Giant Golden Book of Mathematics: Exploring the World of Numbers and Space . Golden Press.
• Crandall, Richard; Pomerance, Carl (2005). Prime Numbers: A Computational Perspective (2nd ed.). Springer. ISBN 0-387-25282-7.
• Dudley, Underwood (1978). “ Section 2: Unique factorization “. Elementary Number Theory (2nd ed.). W.H. Freeman and Co. ISBN 978-0-7167-0076-0.
• “ GIMPS Project Discovers Largest Known Prime Number: 2 82,589,933 -1 “.  Mersenne Research, Inc .
• Ziegler, Günter M. (2004). “The great prime number record races”. Notices of the American Mathematical Society . 51 (4): 414–416.

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‘The Grand Tour’ Penultimate Special Sets Premiere Date

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“ The Grand Tour ” penultimate special has set a premiere date for later this month.

“The Grand Tour: Sand Job,” which sees Jeremy Clarkson , Richard Hammond and James May reunite in Africa, is set to land on Amazon Prime Video on Feb. 16.

“In the remote African country of Mauritania, our trio follow in the footsteps of the legendary Paris-Dakar rally,” reads the logline. “Instead of bespoke Dakar racers, the boys must complete their journey in cheap modified sports cars. Their journey begins with the world’s longest train and sees them tackle the killer Sahara and perilous river crossings, whilst protecting their precious fuel bowser from exploding.”

Popular on Variety

The show’s fifth and final special recently wrapped in Zimbabwe. It is set to launch next year.

Clarkson, whose farming show “Clarkson’s Farm” will continue at Prime Video, has said he decided to take a step back from the automotive series because it is “immensely physical” and he feels “unfit and fat and old.”

“I’ve driven cars higher than anyone else and further north than anyone else,” Clarkson told The Times. “We’ve done everything you can do with a car. When we had meetings about what to do next, people just threw their arms in the air.”

Clarkson, who was widely criticized for a column about Meghan Markle, will next be seen hosting a new series of “Who Wants to Be a Millionaire?,” which premieres Jan. 28 on ITV1.

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Prime Numbers

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A prime number is a natural number greater than 1 that has no positive integer divisors other than 1 and itself. For example, 5 is a prime number because it has no positive divisors other than 1 and 5.

The first 49 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, and 227.

In contrast to prime numbers, a composite number is a positive integer greater than 1 that has more than two positive divisors. For example, 4 is a composite number because it has three positive divisors: 1, 2, and 4. All positive integers greater than 1 are either prime or composite. 1 is the only positive integer that is neither prime nor composite.

Prime numbers are critical for the study of number theory . Nearly all theorems in number theory involve prime numbers or can be traced back to prime numbers in some way. Prime numbers are also important for the study of cryptography . The RSA method of encryption relies upon the factorization of a number into primes. Finally, prime numbers have applications in essentially all areas of mathematics. Prime numbers act as "building blocks" of numbers, and as such, it is important to understand prime numbers to understand how numbers are related to each other.

The Fundamental Theorem of Arithmetic

Identifying prime numbers, primality tests.

• Distribution of Primes

Mersenne Primes

Number theory applications, additional examples.

Main Article: Fundamental Theorem of Arithmetic See Also: Prime Factorization

The fundamental theorem of arithmetic separates positive integers into two classifications: prime or composite.

Fundamental Theorem of Arithmetic Every integer greater than 1 is either prime (it has no divisors other than 1 and itself) or composite (it has more than two divisors). Furthermore, every integer greater than 1 has a unique prime factorization up to the order of the factors.

The prime factorization of a positive integer is that number expressed as a product of powers of prime numbers. Prime factorizations are often referred to as unique up to the order of the factors . This means that each positive integer has a prime factorization that no other positive integer has, and the order of factors in a prime factorization does not matter.

Give the prime factorization of 48. 48 is divisible by the prime numbers 2 and 3. The highest power of 2 that 48 is divisible by is \(16=2^4.\) The highest power of 3 that 48 is divisible by is \(3=3^1.\) Thus, the prime factorization of 48 is \[48=2^4 \times 3^1.\] The fundamental theorem of arithmetic guarantees that no other positive integer has this prime factorization. \(_\square\)

Prime factorization is the primary motivation for studying prime numbers. Many theorems, such as Euler's theorem , require the prime factorization of a number. Prime factorization can help with the computation of GCD and LCM . Prime factorization is also the basis for encryption algorithms such as RSA encryption . In order to develop a prime factorization, one must be able to efficiently and accurately identify prime numbers.

The simplest way to identify prime numbers is to use the process of elimination. List out numbers, eliminate the numbers that have a prime divisor that is not the number itself, and the remaining numbers will be prime. This process can be visualized with the sieve of Eratosthenes .

Sieve of Eratosthenes

Another way to Identify prime numbers is as follows:

• First, choose a number, for example, 119.
• Now, note that prime numbers between 1 and 10 are 2, 3, 5, 7. (Why between 1 and 10? This question is answered in the theorem below.) Divide the chosen number 119 by each of these four numbers.
• If it's divisible by any of the four numbers, then it isn't a prime number; if it's not divisible by any of the four numbers, then it is prime.
• 119 is divisible by 7, so it is not a prime number.

What is the next term in the following sequence?

\[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \ldots \]

Fortunately, one does not need to test the divisibility of each smaller prime to conclude that a number is prime. The number of primes to test in order to sufficiently prove primality is relatively small.

If \(n\) is a composite number, then it must be divisible by a prime \(p\) such that \(p \le \sqrt{n}.\)

Proof by Contradiction : Suppose that \(n\) is a composite number, and it is only divisible by prime numbers that are greater than \(\sqrt{n}.\) Let two of its factors be \(q\) and \(r,\) with \(q,r > \sqrt{n}.\) Then \(n=kqr,\) where \(k\) is a positive integer. However, if \(q\) and \(r\) are both greater than \(\sqrt{n},\) then \(qr>n.\) This cannot be true, because \(n=kqr,\) and \(k\) is a positive integer. Thus, \(n\) must be divisible by a prime that is less than or equal to \(\sqrt{n}.\ _\square\)

One can apply divisibility rules to efficiently check some of the smaller prime numbers. Long division should be used to test larger prime numbers for divisibility. It is helpful to have a list of prime numbers handy in order to know which prime numbers should be tested.

Is 211 a prime number? If 211 is a prime number, then it must not be divisible by a prime that is less than or equal to \(\sqrt{211}.\) \(\sqrt{211}\) is between 14 and 15, so the largest prime number that is less than \(\sqrt{211}\) is 13. It is therefore sufficient to test 2, 3, 5, 7, 11, and 13 for divisibility. 211 is not divisible by any of those numbers, so it must be prime. \(_\square\)

What is the sum of the two largest two-digit prime numbers? If a two-digit number is composite, then it must be divisible by a prime number that is less than or equal to \(\sqrt{100}=10.\) Therefore, it is sufficient to test 2, 3, 5, and 7 for divisibility. Counting backward, 99 is divisible by 3; 98 is divisible by 2; 97 is not divisible by 2, 3, 5, or 7, implying it is the largest two-digit prime number; 96 is divisible by 2; 95 is divisible by 5; 94 is divisible by 2; 93 is divisible by 3; 92 is divisible by 2; 91 is divisible by 7; 90 is divisible by 2; 89 is not divisible by 2, 3, 5, or 7, implying it is the second largest two-digit prime number. The sum of the two largest two-digit prime numbers is \(97+89=186.\) \(_\square\)

What is the largest 3-digit prime number? If a a three-digit number is composite, then it must be divisible by a prime number that is less than or equal to \(\sqrt{1000}.\) \(\sqrt{1000}\) is between 31 and 32, so it is sufficient to test all the prime numbers up to 31 for divisibility. Counting backward, we have the following: 999 is the largest 3-digit number, but as it is divisible by \(3\), it is not prime. 998 is the second largest 3-digit number, but as it is divisible by \(2\), it is not prime. 997 is not divisible by any prime number up to \(31,\) so it must be prime. \(_\square\)

Which prime number follows 1997? 1998 is divisible by 2. If 1999 is composite, then it must be divisible by a prime number that is less than or equal to \(\sqrt{1999}\). \(\sqrt{1999}\) is between 44 and 45, so the possible prime numbers to test are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, and 43. 1999 is not divisible by any of those numbers, so it is prime. \(_\square\)

Sometimes, testing a number for primality does not involve exhaustively searching for prime factors, but instead making some clever observation about the number that leads to a factorization. The next couple of examples demonstrate this.

In the following sequence, how many prime numbers are present? \[121,12321,1234321,123454321,\ldots\] We have \[\begin{align} 121&= 11×11\\ 12321&= 111×111\\ 1234321&= 1111×1111\\ 123454321&= 11111×11111. \end{align}\] So, no numbers in the given sequence are prime numbers. \(_\square\)

How many numbers in the following sequence are prime numbers? \[101,10201,102030201,1020304030201, \ldots\] We have the following factorization: \(101\) has no factors other than 1 and itself. So, it is a prime number. \(10201 = 101×101\). \(102030201 = 10101×10101\). \(1020304030201 = 1010101×1010101\). So, there is only \(1\) prime number in the given sequence. \(_\square\)

Is 12345 a prime number? We have \(\frac{12345}{5}=2469.\) So 12345 is divisible by 5 and therefore is not prime. \(_\square\)

Main Article: Primality Testing

There are other methods that exist for testing the primality of a number without exhaustively testing prime divisors. These methods are called primality tests . One of these primality tests applies Wilson's theorem .

Wilson's Theorem A positive integer \(p>1\) is prime if and only if \[(p-1)! \equiv -1 \pmod{p}.\]

Show that 7 is prime using Wilson's theorem. We have \[\begin{align} 6!&=720\\ 720 &\equiv -1 \pmod{7}. \end{align}\] Therefore, 7 must be prime. \(_\square\)

Testing primes with this theorem is very inefficient, perhaps even more so than testing prime divisors. However, this theorem does give insight that a number's primality is not linked purely to the divisors of that number. There are other "traces" in a number that can indicate whether the number is prime or not.

Most primality tests are probabilistic primality tests . These kinds of tests are designed to either confirm that the number is composite, or to use probability to designate a number as a probable prime . A probable prime is a number that has been tested sufficiently to give a very high probability that it is prime. An example of a probabilistic prime test is the Fermat primality test, which is based on Fermat's little theorem .

Fermat Primality Test Given a positive integer \(n,\) Choose a positive integer \(a>1\) at random that is coprime to \(n\). Compute \(a^{n-1} \bmod {n}.\) If the result is not \(1,\) then \(n\) is composite. Otherwise, \(n\) may or may not be prime. If the result is \(1\) at this step, but \(n\) is composite, then \(n\) is pseudoprime to base \(a.\) Repeat these steps any number of times. Each repetition of these steps improves the probability that the number is prime. However, this process can never guarantee that a number is prime. A composite number that gets a result of \(1\) in the second step for all \(a\) less than \(n\) is called a Carmichael number .

Show that 91 is composite using the Fermat primality test with the base \(a=2\). The goal is to compute \(2^{90}\bmod{91}.\) Use the method of repeated squares . Compute 90 in binary: \[90_{10}=1011010_{2}.\] Compute the residues of the repeated squares of 2: \[\begin{align} 2^{2^0} &\equiv 2 \pmod{91} \\ 2^{2^1} &\equiv 4 \pmod{91} \\ 2^{2^2} &\equiv 16 \pmod{91} \\ 2^{2^3} &\equiv 74 \pmod{91} \\ 2^{2^4} &\equiv 16 \pmod{91} \\ 2^{2^5} &\equiv 74 \pmod{91} \\ 2^{2^6} &\equiv 16 \pmod{91} \\ &\vdots\\ 2^{90} &= 2^{2^6} \times 2^{2^4} \times 2^{2^3} \times 2^{2^1} \\\\ 2^{90} &\equiv (16)(16)(74)(4) \pmod{91} \\ &\equiv 64 \pmod{91}. \end{align}\] The result is not \(1.\) Therefore, \(91\) is not prime. \(_\square\)

This process might seem tedious to do by hand, but a computer could perform these calculations relatively efficiently. Thus, the Fermat primality test is a good method to screen a large list of numbers and eliminate numbers that are composite. Then, a more sophisticated algorithm can be used to screen the prime candidates further.

In general, identifying prime numbers is a very difficult problem. This, along with integer factorization, has no algorithm in polynomial time. In fact, it is so challenging that much of computer cryptography is built around the fact that there is no known computationally feasible way to find the factors of a large number.

Main Article: Distribution of Primes See Also: Infinitely Many Primes

It has been known for a long time that there are infinitely many primes . However, the question of how prime numbers are distributed across the integers is only partially understood. The prime number theorem gives an estimation of the number of primes up to a certain integer.

Prime Number Theorem Let \(\pi(x)\) be the prime counting function . For any real number \(x,\) \(\pi(x)\) gives the number of prime numbers that are less than or equal to \(x.\) Then \[\lim_{x \rightarrow \infty} \frac{\hspace{2mm} \pi(x)\hspace{2mm} }{\frac{x}{\ln{x}}}=1.\] This implies that for sufficiently large \(x,\) \[\pi(x) \sim \frac{x}{\ln{x}}.\]

There are many open questions about prime gaps. A prime gap is the difference between two consecutive primes. For example, the prime gap between 13 and 17 is 4. Bertrand's postulate gives a maximum prime gap for any given prime.

Bertrand's Postulate For any integer \(n>3,\) there always exists at least one prime number \(p\) such that \[n<p<2n-2.\] This implies that for the \(k^\text{th}\) prime number, \(p_k,\) the next consecutive prime number is subject to \[p_{k+1}<2p_k.\]

This is, unfortunately, a very weak bound for the maximal prime gap between primes. Prime gaps tend to be much smaller, proportional to the primes. For example, the first occurrence of a prime gap of at least 100 occurs after the prime 370261 (the next prime is 370373, a prime gap of 112).

The most famous problem regarding prime gaps is the twin prime conjecture . This conjecture states that there are infinitely many pairs of primes for which the prime gap is 2, but as of this writing, no proof has been discovered.

Another famous open problem related to the distribution of primes is the Goldbach conjecture . This conjecture states that every even integer greater than 2 can be expressed as the sum of two primes.

One of the most significant open problems related to the distribution of prime numbers is the Riemann hypothesis . Although the Riemann hypothesis has wide-reaching implications in number theory, Riemann's original motivation for formulating the conjecture was to better understand the distribution of prime numbers. The Riemann hypothesis relates the real parts of the zeros of the Riemann zeta function to the oscillations of the prime numbers about their "expected" positions given the estimation of the prime counting function above.

Main Article: Mersenne Primes

A Mersenne prime is a prime that can be expressed as \(2^p-1,\) where \(p\) is a prime number.

The first five Mersenne primes are listed below:

\[\begin{array}{c|rr} p & 2^p-1= & M_p\\ \hline 2 & 2^2-1= & 3 \\ 3 & 2^3-1= & 7 \\ 5 & 2^5-1= & 31 \\ 7 & 2^7-1= & 127 \\ 13 & 2^{13}-1= & 8191 \end{array}\]

Note that having the form of \(2^p-1\) does not guarantee that the number is prime. \(2^{11}-1=2047\) is not a prime number; its prime factorization is \(23 \times 89.\)

From the list above, it might seem as though Mersenne primes are relatively easy to find by simply plugging in prime numbers into \(2^p-1\). However, Mersenne primes are exceedingly rare. As of January 2018, only 50 Mersenne primes are known, the largest of which is \(2^{77,232,917}-1\). This number is also the largest known prime number. In fact, many of the largest known prime numbers are Mersenne primes. This is due to the Lucas-Lehmer primality test , which is an efficient algorithm that is specific to testing primes of the form \(2^p-1\). Although Mersenne primes continue to be discovered, it is an open problem whether or not there are an infinite number of them.

Another notable property of Mersenne primes is that they are related to the set of perfect numbers . A perfect number is a positive integer that is equal to the sum of its proper positive divisors. Each Mersenne prime corresponds to an even perfect number:

Let \(M_p\) be a Mersenne prime. Then \(\frac{M_p\big(M_p+1\big)}{2}\) is an even perfect number. Furthermore, all even perfect numbers have this form.

Give the perfect number that corresponds to the Mersenne prime 31. The perfect number is given by the formula above: \[\frac{(31)(32)}{2}=496.\] This number can be shown to be a perfect number by finding its prime factorization: \[496=2^4\times 31.\] Then listing out its proper divisors gives \[\text{proper divisors of 496}=\{1,2,4,8,16,31,62,124,248\}.\] Summing these divisors, we have \[1+2+4+8+16+31+62+124+248=496.\ _\square\]

One of the most fundamental theorems about prime numbers is Euclid's lemma.

Euclid's Lemma Let \(p\) be prime. If \(p \mid ab\), then \(p \mid a\) or \(p \mid b\).

By Contradiction: Suppose \(p\) does not divide \(a\). Since the only divisors of \(p\) are \(1\) and \(p,\) and \(p\) doesn't divide \(a,\) we must have \(\gcd (a, p) =1.\) By Bezout's identity , there exist some \(u\) and \(v\) such that \(ua+vp=1\). Multiplying both sides of this equation by \(b\) gives \(b=uab+vpb\). Now \(p\) divides \(uab\) \((\)since it is given that \(p \mid ab),\) and \(p\) also divides \(vpb\). Therefore, \(p\) divides their sum, which is \(b\). Without loss of generality, if \(p\) does not divide \(b,\) then it must divide \(a.\) \( _\square \)

Euclid's lemma can seem innocuous, but it is incredibly important for many proofs in number theory. For example, it is used in the proof that the square root of 2 is irrational .

Prime factorizations can be used to compute GCD and LCM .

GCD and LCM using Prime Factorizations Given positive integers \(m\) and \(n,\) let their prime factorizations be given by \[\begin{align} m&=p_1^{j_1} \times p_2^{j_2} \times p_3^{j_3} \times \cdots\\ n&=p_1^{k_1} \times p_2^{k_2} \times p_3^{k_3} \times \cdots, \end{align}\] where \(p_1, p_2, p_3, \ldots\) are distinct primes and each \(j_i\) and \(k_i\) are integers. Then the GCD of these integers is given by \[\gcd(m,n)=p_1^{\min(j_1,k_1)} \times p_2^{\min(j_2,k_2)} \times p_3^{\min(j_3,k_3)} \times \cdots,\] and the LCM of these integers is given by \[\text{lcm}(m,n)=p_1^{\max(j_1,k_1)} \times p_2^{\max(j_2,k_2)} \times p_3^{\max(j_3,k_3)} \times \cdots.\]

Using prime factorizations, what are the GCD and LCM of 36 and 48? The prime factorizations are \[\begin{align} 36 &= 2^2 \times 3^2 \\ 48 &= 2^4 \times 3^1. \end{align}\] Each number has the same primes, 2 and 3, in its prime factorization. The GCD is given by taking the minimum power for each prime number: \[\begin{align} \gcd(36,48) &= 2^{\min(2,4)} \times 3^{\min(2,1)} \\ &= 2^2 \times 3^1 \\ &= 12. \end{align}\] The LCM is given by taking the maximum power for each prime number: \[\begin{align} \text{lcm}(36,48) &= 2^{\max(2,4)} \times 3^{\max(2,1)} \\ &= 2^4 \times 3^2 \\ &= 144.\ _\square \end{align}\]

Prime numbers are important for Euler's totient function .

Given a positive integer \(n\), Euler's totient function , denoted by \(\phi(n),\) gives the number of positive integers less than \(n\) that are co-prime to \(n.\)

What is \(\phi(10)?\) Listing out the positive integers that are less than 10 gives \[\{1,2,3,4,5,6,7,8,9\}.\] Of those numbers, list the subset of numbers that are co-prime to 10: \[\{1,3,7,9\}.\] This set contains 4 elements. Therefore, \(\phi(10)=4.\ _\square\)

If \(n\) is a power of a prime, then Euler's totient function can be computed efficiently using the following theorem:

For any given prime \(p\) and positive integer \(n\), \[\phi(p^n)=p^n-p^{n-1}.\]

Then, the value of the function for products of coprime integers can be computed with the following theorem:

Given co-prime positive integers \(m\) and \(n\), \[\phi(mn)=\phi(m) \times \phi(n).\]

The consequence of these two theorems is that the value of Euler's totient function can be computed efficiently for any positive integer, given that integer's prime factorization.

What is \(\phi(48)?\) The prime factorization of 48 is \[48=2^4 \times 3^1.\] Then, \[\begin{align} \phi(2^4) &= 2^4-2^3=8 \\ \phi(3^1) &= 3^1-3^0=2 \\ &\vdots\\ \phi(48) &= 8 \times 2=16.\ _\square \end{align}\]

Euler's totient function is critical for Euler's theorem .

Euler's Theorem Let \(a\) and \(n\) be coprime integers with \(n>0\). Then, \[a^{\phi(n)} \equiv 1 \pmod{n}.\]

If \(n\) is a prime number, then this gives Fermat's little theorem .

Fermat's Little Theorem Let \(p\) be a prime number and let \(a\) be an integer coprime to \(p.\) Then, \[a^{p-1} \equiv 1 \pmod{p}.\]

The properties of prime numbers can show up in miscellaneous proofs in number theory .

Let \(p\) be a prime number greater than \(3\). Prove that \(p^2-1\) is always divisible by 6. Any integer can be written in the form \(6k+n,\ n \in \{0,1,2,3,4,5\}\). \(6k\) cannot be a prime. \(6k+1\) may be a prime. \(6k+2\) cannot be a prime. \(6k+3\) cannot be a prime. \(6k+4\) cannot be a prime. \(6k+5\) may be a prime. Thus, any prime \(p > 3\) can be represented in the form \(6k+5\) or \(6k+1\). \(p^2-1\) can be factored to \((p+1)(p-1).\) Case 1 : \(p=6k+1\) If this is the case, \(p^2-1=(6k+2)(6k),\) which implies \(6 \mid (p^2-1).\) Case 2 : \(p=6k+5\) If this is the case, \(p^2-1=(6k+6)(6k+4),\) which implies \(6 \mid (p^2-1).\) One of the factors, \(p-1\) or \(p+1\), will be divisible by \(6\). Thus, \(p^2-1\) is always divisible by \(6\). \(_\square\)

\(n\) is the greatest prime number less than 50, and \(m\) is the least prime number greater than 50. What is the value of \(n+m?\) Let's work backward for \(n\). \(49\) is divisible by \(7\), and from the property of primes it is enough information to conclude that the number is not prime. \(48\) is divisible by \(2,\) so cancel it. When we look at \(47,\) it doesn't have any divisor other than one and itself. So it is indeed a prime: \(n=47.\) We use the same process in looking for \(m\). \(51\) is divisible by \(3\). \(52\) is divisible by \(2\). \(53\) doesn't have any other divisor other than one and itself, so it is indeed a prime: \(m=53.\) Therefore, \[n+m=100.\ _\square\]

Consider the expression \(2^{n}-1,\) where \(n\) is an integer greater than \(1.\) What are the least two values of \(n\) for which the expression doesn't produce a prime number? Let's check by plugging in numbers in increasing order. \(2^{2}-1=3\) is prime. \(2^{3}-1=7\) is prime. \(2^{4}-1=15\), which is divisible by 3, so it isn't prime. \(2^{5}-1=31\) is prime. \(2^{6}-1=63\), which is divisible by 7, so it isn't prime. Therefore, the least two values of \(n\) are 4 and 6. \(_\square\)

An emirp (prime spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. This definition excludes the related palindromic primes. The term reversible prime may be used to mean the same as emirp, but may also, ambiguously, include the palindromic primes.

The sequence of emirps begins 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991, ... (sequence A006567 in the OEIS).

All non-palindromic permutable primes are emirps.

As of November 2009, the largest known emirp is 1010006+941992101×104999+1, found by Jens Kruse Andersen in October 2007.

The term 'emirpimes' (singular) is used also in places to treat semiprimes in a similar way. That is, an emirpimes is a semiprime that is also a (distinct) semiprime upon reversing its digits.

• Prime Factorization
• Divisibility Rules
• Fundamental Theorem of Arithmetic
• Chinese Remainder Theorem
• Mersenne Prime
• RSA Encryption

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Prime Numbers Chart and Calculator

A Prime Number is:

a whole number above 1 that cannot be made by multiplying other whole numbers

(if we can make it by multiplying other whole numbers it is a Composite Number )

Here we see it in action:

Here is a list of all the prime numbers up to 1,000:

• Is 8 a Prime Number? No , because it can be made by 2×4=8
• Is 73 a Prime Number? Yes , as no other whole numbers multiply together to make it.

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How to watch the 2024 PGA Championship today

A fter a dramatic start to the 2024 PGA Championship, including an early morning arrest Friday for top-ranked Scottie Scheffler , the final two days of the PGA Championship are here. Xander Schauffele comes into the weekend on top of the leaderboard , with Collin Morikawa, Sahith Theegala and Scheffler close behind.

Keep reading to find out how to watch all of this weekend's action at the 2024 PGA Championship, live from Louisville, KY. 

CBS Essentials, CBS and Paramount+ are all subsidiaries of Paramount. CBS is one of the broadcast homes of the 2024 PGA Championship.

PGA Championship: When and how to watch Round 3 and Round 4

Round 3 of the PGA Tournament will be played on Saturday, May 18, 2024. Round 4 will be played on Sunday, May 19, 2024.

Competition both days will air live on ESPN and CBS, with ESPN's coverage going from 10:00 a.m. - 1:00 p.m. ET. Coverage continues on CBS from 1:00 p.m. to 7:00 p.m. ET.  You can watch afternoon coverage on  Paramount+ with Showtime , watch all of the weekend's events on ESPN+ or one of the streaming platforms featured below.

How to watch the PGA Championship without cable

If you're watching the PGA Championship without cable, you can still tune in. It's easy (and via some providers, free). 

PGA Championship afternoon coverage: Paramount+ with Showtime

CBS will host some of this year's PGA Championship, which means  Paramount+ with Showtime  subscribers can stream CBS-aired PGA golf live.

The streamer offers access to all PGA golf locally and nationally televised on CBS on its Paramount + with Showtime tier. In addition, you can watch top-tier soccer like the Champions League live and SEC college football games, plus popular shows such as "Survivor" and "NCIS" and NFL games airing on CBS next season

A subscription to Paramount+ with Showtime is $12 per month. The streamer offers a seven-day free trial. (You won't be able to stream PGA golf live with a Paramount+ Essential subscription.)

Start watching Paramount + free on Amazon Prime Video

Paramount+ content, including live sports, is available to stream on  Amazon Prime  via a Paramount+ on Prime Video add-on subscription. Prime Video also carries some of the best sports docs, including "Kelce," which documents Philadelphia Eagles star Jason Kelce's last season in the league.

Amazon is offering a seven-day free trial of  Paramount+ . Tap the button below to start your trial and start streaming now for free. After the free trial period ends, a subscription to Paramount+ Essential is $6 per month, while the Paramount+ with Showtime tier is $12 per month.

Watch the 2024 PGA Championship for free with Fubo

You can also catch PGA golf on  Fubo . Fubo is a sports-centric streaming service that offers access to network-aired sports like the 2024 PGA Championship and almost every NFL game next season. Packages include the live feed of sports and programming airing on CBS, Fox, NBC, ESPN, the NFL Network and more, so you'll be able to watch more than just this weekend's tournament -- all without a cable subscription.

To watch the 2024 PGA Championship without cable, start a seven-day free trial of Fubo . You can begin watching immediately on your TV, phone, tablet or computer. In addition to PGA Tour golf, you'll have access to NFL football, Fubo offers NCAA college sports , MLB, NBA , NHL, MLS and international soccer games. Fubo's Pro Tier is priced at $80 per month after your free seven-day trial. 

Top features of Fubo Pro Tier:

• There are no contracts with Fubo -- you can cancel at any time.
• The Pro tier includes over 180 channels, including NFL Network and Golf Channel.
• Fubo includes all the channels you'll need to watch college and pro sports, including CBS (not available through Sling TV).
• All tiers come with 1,000 hours of cloud-based DVR recording.
• Stream on your TV, phone, and other devices.

Stream the PGA Championship on Sling TV for half price

If you don't have cable TV that includes ESPN, one of the most cost-effective ways to stream PGA golf is through a subscription to Sling TV. The streamer also offers access to the PGA golf airing on ESPN, plus top-tier sports channels like NFL Network, local NBC, Fox and ABC affiliates (where available) and ESPN with its Orange + Blue Tier plan. Also worth noting: Sling TV comes with 50 hours of cloud-based DVR recording space included, perfect for recording all the season's top NFL matchups.

That plan normally costs $60 per month, but the streamer is currently offering a 50% off promotion for your first month, so you'll pay just $30. You can learn more by tapping the button below.

Note: Because Sling TV doesn't carry CBS, you won't be able to watch CBS-aired programming like the PGA Championship or next year's NFL games on CBS. To watch these games, plus PGA golf, UEFA Champions League and more live sports, we recommend you also  subscribe to Paramount+ with Showtime . Paramount+ with Showtime costs $12 per month after a one-week free trial.

Top features of Sling TV Orange + Blue tier:

• There are 46 channels to watch in total, including local ESPN, Fox and ABC affiliates (where available).
• You get access to most local NFL games and nationally broadcast games at the lowest price.
• All subscription tiers include 50 hours of cloud-based DVR storage.
• You can add Golf Channel, NBA TV, NHL Network, NFL RedZone, MLB Network, Tennis Channel and more sports-oriented channels (25 in total) via Sling TV's $15 per month Sports Extras add-on.

Watch the PGA Championship on Hulu + Live TV

You can watch the 2024 PGA Championship and many other top-tier sports, including NFL Network, with  Hulu + Live TV . The bundle features access to 90 channels, including both ESPN and CBS. Unlimited DVR storage is also included. Watch every PGA event on every network with Hulu + Live TV, plus catch live NFL games next season, exclusive live regular season games, popular studio shows (including NFL Total Access and the Emmy-nominated show Good Morning Football) and lots more.

Hulu + Live TV comes bundled with ESPN+ and Disney+ for $77 per month.

Stream the PGA Championship on ESPN+

Stream both days' coverage of the PGA Championship on ESPN+ , which offers exclusive live sports, original shows, and a vast library of on-demand content, including the entire 30 For 30 series and more. 

You can stream ESPN+ through the ESPN app on your favorite  mobile device  and ESPN.com. It's also available as an app through major smart TV streaming platforms and  gaming consoles such as the PS5 . It is important to note that ESPN+ does not include access to the ESPN network. It is a separate sports-centric service, with separate sports programming. 

An ESPN+ subscription costs $11 per month, or save 15% when you pay annually ($110).  ESPN+ is also currently offering a cost-saving bundle. Get ESPN+ (with ads), Disney+ (with ads) and Hulu (with ads) for $15 per month

Here's a sampling of what's available on ESPN+:

• Exclusive fantasy sports tools and content from some of the sports world's most respected voices in sports.  
• Select WNBA games, including Caitlin Clark's WNBA regular season debut.
• Every Fight Night UFC event UFC PPV event (PPV events are subject to an additional charge).
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• College sports  including the Ivy League, Big Sky Conference and Atlantic A10 Conference.
• MLB and the World Series.
• Top-tier tennis including the Australian Open and Wimbledon.
• The PGA Tour and the  Masters .

Watch the 2024 PGA Championship live on CBS with a digital HDTV antenna

You can also watch championship golf on TV with an affordable indoor antenna, which pulls in local over-the-air HDTV channels such as CBS, NBC, ABC, Fox, PBS, Univision and more. Here's the kicker: There's no monthly charge.

For anyone living in a partially blocked-off area (those near mountains or first-floor apartments), a  digital TV antenna  may not pick up a good signal -- or any signal at all. But for many homes, a digital TV antenna provides a seriously inexpensive way to watch NBA basketball without paying a cable company. Indoor TV antennas can also provide some much-needed TV backup if a storm knocks out your cable.

This amplified digital antenna with a 50-mile range can receive hundreds of HDTV channels, including ABC, CBS, NBC, PBS, Fox and Univision and can filter out cellular and FM signals. It delivers a high-quality picture in 1080p HDTV and top-tier sound.

Stream the 2024 PGA Championship on ESPN+

ESPN+ is ESPN's subscription streaming platform, which offers PGA Tour golf, exclusive live events, original studio shows and top-tier series that aren't accessible on the ESPN networks. ESPN+ subscribers may purchase UFC PPV events and access the platform's vast archive of on-demand content, including the entire "30 For 30" catalog, game replays and select ESPN films. 

It is important to note that ESPN+ does not include access to the ESPN network. It is a separate sports-centric service, with separate sports programming.

ESPN+ offers exclusive live sports, original shows, a vast library of on-demand content, including the entire "30 For 30" series and more. ESPN+ costs $10.99 per month; it can be bundled with Hulu and Disney+ for $14.99 per month.

Here's a sampling of what's available on ESPN+.

• College sports including the Ivy League, Big Sky Conference and Atlantic A10 Conference.
• The PGA Tour and the Masters.

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2024 PGA Championship live stream, where to watch: TV coverage, channel, Round 4 schedule, tee times

Watch every single shot from the biggest names in golf throughout the final round at valhalla.

The final round of the 2024 PGA Championship is underway as Valhalla Golf Club offers an opportunity for some of the best golfers in the world to streak up the leaderboard and claim the second major championship of the year. The Wanamaker Trophy and $3.3 million -- the winner's share out of the PGA Championship purse of $18.5 million -- are on the line for the victor, but it will certainly not be a walk in the park despite Valhalla offering plenty of low numbers on the week.

Follow the action all day Sunday as CBS Sports offers  PGA Championship live leaderboard coverage and updates  with analysis and highlights throughout the round.

Round 4 should be electric with an absolutely stacked leaderboard battling down the stretch . Olympic gold medalist Xander Schauffele and two-time major winner Collin Morikawa began the round out in front as the co-leaders, but a ton of big-time names were on their tails to start the final round, including Bryson DeChambeau, Viktor Hovland, Sahith Theegala and Justin Rose. Then there were Justin Thomas, Rory McIlroy and Jordan Spieth, who certainly had some work to do but nevertheless remained just on the outskirts of competing for the Wanamaker.

World No. 1 Scottie Scheffler, who was  arrested outside Valhalla on Friday for disobeying a police officer in a traffic-related incident, is not among those in striking position any longer as his chaotic Friday gave way to an ineffective Saturday in which he posted an over-par round for the first time since August 2023 .

Given how much talent in the field, whoever ultimately prevails at Valhalla will surely have earned the Wanamaker and a winner's share that is the largest in PGA Championship history.

There is so much to follow Sunday that it can be overwhelming at times. No worries, though, we have you covered.

CBS Sports is offering live coverage of the 2024 PGA Championship from start to finish Sunday along with live streaming of the tournament. Be sure to check out our complete PGA Championship TV schedule and coverage guide so you can follow along.

All times Eastern

Round 4 -- Sunday, May 19

Round 4 start time:  7:45 a.m. [ Tee times ]

PGA Championship live stream:  8-10 a.m. on ESPN+

• Featured Groups -- 8 a.m.
• Holes 16-18 -- 8 a.m.

Early TV coverage:  10 a.m. to 1 p.m. on ESPN,  fubo  (Try for free)

Prime TV coverage:  1-7 p.m. on CBS, Paramount+

TV simulcast live stream:  1-7 p.m. Desktop and mobile:  Free on  CBSSports.com ,  CBS Sports App Connected devices*:  Available on  Paramount+ ,  CBS Sports App *Paramount+ login required

Clubhouse Report:  8-9 p.m. on CBS Sports Network Round 4 encore:  9 p.m. to 3 a.m. on CBS Sports Network

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No one is catching Scottie Scheffler anytime soon, but Xander Schauffele’s victory in the PGA Championship shuffled the standings near the top of the FedExCup and Official World Golf Ranking.

Schauffele rose to a career-best No. 2 in the world ranking and remained second in the FedExCup standings Sunday, lessening the gap between him and Scheffler.

With the 750 points he earned Sunday, Schauffele now trails Scheffler by 1,417 points. Scheffler earned 191 points with his T8 finish at Valhalla Golf Club.

The fact that Schauffele was second in the FedExCup before picking up his first win of the season is a testament to his consistency in 2024. Schauffele had seven top-10s in 11 starts before arriving at Valhalla, including runners-up at THE PLAYERS and last week’s Wells Fargo Championship, where he was overtaken by Rory McIlroy in the final round.

Schauffele surpassed McIlroy in the world ranking with Sunday’s win, moving up one spot after his first major victory. Schauffele, who played his first PGA TOUR season in 2017, entered the top 10 in the OWGR in December 2018 and cracked the top 5 three years later. He became the world’s third-ranked player for the first time after finishing eighth at the Masters.

PGA Championship

Valhalla golf club.

Payday at Valhalla

PGA Championship 2024: Prize money payouts at Valhalla hit another record-breaking number

It’s a sign of the times in men’s professional golf. The question wasn’t whether the PGA of America was going to increase its prize money payout for each golfer at this year’s PGA Championship to another record payday. Rather, it was by how much.

That amount turned out to be $1 million, the PGA of America announcing Saturday that its overall purse had jumped to $18.5 million. On Sunday, Xander Schauffele birdied the 72nd hole to shoot 65 at Valhalla Golf Club in Louisville to take home $3.33 million.

To put that number in perspective, just four years ago at TPC Harding Park, the overall purse was $11 million. But then the PGA of America bumped the amount by $1 million for the 2021 PGA at Kiawah, $3 million more for the 2022 PGA at Southern Hills and another $2.5 million for the 2023 PGA at Oak Hill.

Here's a look at the evolution of the prize money payout in the PGA Championship, offering a revealing glance at the trajectory of golf purses in general over the years:

Year: Winner's Pay, Total Purse 1916: $500, $2,580 (first year of the event) 1931: $1,000, $7,200 (first year winner's pay increased) 1953: $5,000, $20,700 (first year winner's pay was $5K) 1958: $5,500, $39,388 (first year of stroke play; winner's amount actually decreased that year) 1965: $25,000, $149,700 1978: $50,000, $300,240 1983: $100,000, $608,099 1988: $160,000, $1,000,000 (first year with a $1M total purse) 1993: $300,000, $1,702,750 1998: $540,000, $2,886,800 2000: $900,000, $5,031,100 (first year with a $5M total purse) 2003: $1,080,000, $5,938,300 (first year with $1M-plus to the winner) 2009: $1,350,000, $7,484,500 2014: $1,800,000, $9,913,000 2018: $1,980,000, $11,000,000 2021: $2,160,000, $12,000,000 2022: $2,700,000, $15,000,000 2023: $3,150,000, $17,500,000

The latest jump, however, leaves the PGA Championship’s purse shy of the $20 million paydays offered in the PGA Tour’s new signature events as well as the $20 million paydays being played for individually on the LIV Golf circuit.

Here is a breakdown of the prize money payouts for each place in the field.

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Win: Xander Schauffele, -21//263, $3,330,000

2: Bryson DeChambeau, -20/264, $1,998,000

3: Viktor Hovland, -18/266, $1,258,000

T-4: Thomas Detry, -15/269, $814,000

T-4: Collin Morikawa, -15/269, $814,000

T-6: Shane Lowry, -14/270, $639,440

T-6: Justin Rose, -14/270, $639,440

T-8: Robert MacIntyre, -13/271, $521,417.50

T-8 : Justin Thomas, -13/271, $ 521,417.50

T-8 : Scottie Scheffler, -13/271, $ 521,417.50

T-8 : Billy Horschel, -13/271, $ 521,417.50

T-12: Dean Burmester, -12/272, $359,943.33

T-12 : Lee Hodges, -12/272, $ 359,943.33

T-12 : Taylor Moore, -12/272, $ 359,943.33

T-12 : Rory McIlroy, -12/272, $ 359,943.33

T-12 : Alex Noren, -12/272, $ 359,943.33

T-12 : Sahith Theegala, -12/272, $ 359,943.33

T-18: Tony Finau, -11/273, $230,764

T-18 : Harris English, -11/273, $ 230,764

T-18 : Austin Eckroat, -11/273, $ 230,764

T-18 : Keegan Bradley, -11/273, $ 230,764

T-18 : Ryo Hisatsune, -11/273, $ 230,764

T-23: Russell Henley, -10/274, $170,136.67

T-23: Tom Hoge, -10/274, $ 170,136.67

T-23: Maverick McNealy, -10/274, $ 170,136.67

T-26: Tom Kim, -9/275, $113,962.22

T-26 : Mark Hubbard, -9/275, $ 113,962.22

T-26 : Ben Kohles, -9/275, $ 113,962.22

T-26 : Kurt Kitayama, -9/275, $ 113,962.22

T-26 : Min Woo Lee, -9/275, $ 113,962.22

T-26 : Brian Harman, -9/275, $ 113,962.22

T-26 : Corey Conners, -9/275, $ 113,962.22

T-26 : Brooks Koepka, -9/275, $ 113,962.22

T-26 : Tommy Fleetwood, -9/275, $ 113,962.22

T-35: Hideki Matsuyama, -8/276, $79,182.50

T-35: Max Homa, -8/276, $ 79,182.50

T-35: Doug Ghim, -8/276, $ 79,182.50

T-35: Brice Garnett, -8/276, $ 79,182.50

T-39: Aaron Rai, -7/277, $66,847.50

T-39 : Alexander Bjork, -7/277, $ 66,847.50

T-39 : Joaquin Niemann, -7/277, $ 66,847.50

T-39 : Jordan Smith, -7/277, $ 66,847.50

T-43: Jordan Spieth, -6/278, $48,969

T-43 : Matt Wallace, -6/278, $ 48,969

T-43 : Jason Day, -6/278, $ 48,969

T-43 : Will Zalatoris, -6/278, $ 48,969

T-43 : Lucas Glover, -6/278, $ 48,969

T-43 : Adam Svensson, -6/278, $ 48,969

T-43 : Byeong Hun An, -6/278, $ 48,969

T-43 : Grayson Murray, -6/278, $ 48,969

T-43 : Dustin Johnson, -6/278, $ 48,969

T-43 : Lucas Herbert, -6/278, $ 48,969

T-53: Zac Blair, -5/279, $32,587.14

T-53 : Thorbjorn Olesen, -5/279, $ 32,587.14

T-53 : Patrick Reed, -5/279, $ 32,587.14

T-53 : Jeper Svensson, -5/279, $ 32,587.14

T-53 : Patrick Cantlay, -5/279, $ 32,587.14

T-53 : Erik van Rooyen, -5/279, $ 32,587.14

T-53 : Andrew Putnam, -5/279, $ 32,587.14

T-60: Gary Woodland, -4/280, $27,016.67

T-60: Adam Hadwin, -4/280, $ 27,016.67

T-60: Talor Gooch, -4/280, $ 27,016.67

T-63: Cameron Smith, -3/281, $26,000

T-63 : Tyrrell Hatton, -3/281, $25,202

T-63 : Cameron Young, -3/281, $25,202

T-63 : Rickie Fowler, -3/281, $25,202

T-63 : S.H. Kim, -3/281, $25,202

T-68: Nicolai Hojgaard, -2/282, $23,537.50

T-68 : Luke Donald, -2/282, $ 23,537.50

T-68 : Rasmus Hojgaard, -2/282, $ 23,537.50

T-68 : Sebastian Soderberg, -2/282, $ 23,537.50

72: Braden Shattuck (PGA pro), -1/283, $22,830

T-73: Martin Kaymer, E/284, $22,560

T-73: Alejandro Tosti, E/284, $22,560

75: Ryan Fox, +2/286, $22,350

76: Stephan Jaeger, +5/289, $22,230

77: Jeremy Wells (PGA pro), +6/290, $22,140

78: Brendon Todd, +9/293, $22,100

Players missing the cut and turning in a 36-hole score were paid $4,000 each. Any player making the cut but failing to submit a 72-hole score were also be paid $4,000.

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Python Program to Check Prime Number

To understand this example, you should have the knowledge of the following Python programming topics:

A positive integer greater than 1 which has no other factors except 1 and the number itself is called a prime number. 2, 3, 5, 7 etc. are prime numbers as they do not have any other factors. But 6 is not prime (it is composite) since, 2 x 3 = 6 .

Example 1: Using a flag variable

In this program, we have checked if num is prime or not. Numbers less than or equal to 1 are not prime numbers. Hence, we only proceed if the num is greater than 1.

We check if num is exactly divisible by any number from 2 to num - 1 . If we find a factor in that range, the number is not prime, so we set flag to True and break out of the loop.

Outside the loop, we check if flag is True or False .

• If it is True , num is not a prime number.
• If it is False , num is a prime number.

Note : We can improve our program by decreasing the range of numbers where we look for factors.

In the above program, our search range is from 2 to num - 1 .

We could have used the range, range(2,num//2) or range(2,math.floor(math.sqrt(num)+1)) . The latter range is based on the fact that a composite number must have a factor less than or equal to the square root of that number. Otherwise, the number is prime.

You can change the value of variable num in the above source code to check whether a number is prime or not for other integers.

In Python, we can also use the for...else statement to do this task without using an additional flag variable.

Example 2: Using a for...else statement

Here, we have used a for..else statement to check if num is prime.

It works on the logic that the else clause of the for loop runs if and only if we don't break out the for loop. That condition is met only when no factors are found, which means that the given number is prime.

So, in the else clause, we print that the number is prime.

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Live from Paris: Taylor Swift adds 'Tortured Poets' songs to Eras Tour in France

PARIS, France — The Chairman jump-started her tour after a two-month hiatus, and anticipation was high over whether Taylor Swift would incorporate her 11th era album into the Eras Tour set at La Défense Arena on Thursday.

She did and then some.

Swift extended the set list by one track to land at 46 songs. But many changes were made along the way. She added a new era for "The Tortured Poets Department," cut a handful of songs and rearranged most of the eras including combining the sets for "Folklore" and "Evermore."

She also debuted a number of new outfits along the way.

Delivering all the updates were Bryan West, the USA TODAY Network's full-time Taylor Swift reporter, from inside La Défense Arena in Paris, with Swiftie and trending reporter Anthony Robledo assisting from his desk.

Need a break? Play the USA TODAY Daily Crossword Puzzle.

We hereby present the minutes from the first Eras Tour meeting in Paris, France.

More: Taylor Swift unveils new Eras Tour set list in Paris

'Midnights' set follows secret songs

Swift stuck to her original set list for the "Midnights" era by singing "Lavender Haze," "Anti-Hero" and "Midnight Rain." She ended the night with a triumphant rendition of "Karma" as usual.

'loml' is second secret song

Fans got one more taste of "The Tortured Poets Department" when Swift played the heartbreaking song "loml" on the piano during her secret set.

'Paris' is a secret song, of course

Swift started her secret set with "Paris," a bonus song from her "Midnights" album.

'Tortured Poets' set list

Swift played these songs from her newest album:

• "But Daddy I Love Him"
• "So High School"
• "Who's Afraid of Little Old Me"
• "Fortnight"
• "The Smallest Man Who Ever Lived"
• "I Can Do It with a Broken Heart"

New 'Tortured Poets' set is here

Toward the end of her show, Swift officially launched the long-awaited “The Tortured Poets Department” section of the tour after the 1989 era.

Swift performed the track “But Daddy I Love Him” off the April 19 album for her ecstatic fans while wearing a new beige dress with cursive letters.  

A road spiraled down the catwalk in an animation to initiate the era. The background animation showed a deserted road while large cages animated and fell on the road.

“Oh my god you should see your faces,” she sang.

She quickly segued into a brief moment of “So High School,” the beloved track speculated to be about her romance with Kansas City Chiefs tight end Travis Kelce. She quickly performed the powerhouse anthem “Who’s Afraid of Little Old Me.”

Swift switched to “Down Bad” before seamlessly transitioning into the lead single “Fortnight.” Stage production recreated an asylum that resembled the music video for the song with Post Malone, recorded harmony played concluding the song. 

Dancers entered the stage in a marching band style format playing drums as she performed the wrath-fueled song “The Smallest Man Who Ever Lived.” Jan Ravnik and Kameron Saunders then revived Swift in a jazzy skit to set up “I Can Do With A Broken Heart,” where she debuted another new outfit. 

“The End” text displayed wrapping up the era that’s likely here to stay. Other beloved tracks like “Guilty As Sin” and “Florida!” did not make the cut.

New '1989' outfit

Swift never goes out of Style. During her “1989” set, she debuted a new, sequined outfit of blue and pink. Her sparkly shoes matched with a different colored heel on each. For the bridge of “Blank Space,” the crowd started chanting in Sydney, Australia, “Syd-ney” as a nod to the 1989 World Tour concert movie where Swift would chant the name of the city she was in after hitting a pole with a club two times. Singapore followed the chant and Taylor Nation tweeted they want every city to incorporate the fan-fueled chant.In Paris, thousands chanted “Par-ee,” the French way to say the capital city.

Swift made no cuts to the "1989" set.

Four songs cut from 'Folklore' and 'Evermore'

The new set list cut four tracks from the "Folklore" and "Evermore" eras:

• "'Tis the Damn Season"
• "Tolerate It"
• "The Last Great American Dynasty"

The combined set included eight songs altogether and wrapped up with "Willow."

'Folkmore' set debuts

Swift revealed she merged the 2020 pandemic albums “Folklore” and “Evermore” into one section. She told fans she wrote “Folklore” to represent spring and summer with “Evermore” representing fall and winter in what she always considered sister albums.

“On the Eras Tour we have now reunited the sisters, combined them into one chapter,” she said. “You can call it whatever you want as long as you promise to sing ‘Champagne Problems’ with me.”

'Folklore' set: 'Cardigan' makes the cut

Many fans had speculated "Cardigan" would be cut from the set list. But Swift actually moved it to the top of the "Folklore" set.

Up next: 'Reputation'

Swift kicked off her fifth set with "Ready for It?"

Another surprise: 'Speak Now' era comes fourth

Swift updated her "Speak Now" set with an intro dance featuring Raphael Thomas and eight other dancers wearing flowy purple dresses. She then sang "Enchanted."

Fan favorite track "Long Live" was cut from the set list.

Order of eras changed

After a fairly routine "Fearless" set, Swift altered her concert by including the “Red” era third instead of the expected “Evermore” set. Previously, "Red" came fifth.

With a new shirt that read “This Is Not Taylor’s Version,” the singer performed "22" and showed her audience that they needed to leave any expectations at home. 

The relocated "Red" set included the same four songs: "22," "We Are Never Ever Getting Back Together," "I Knew You Were Trouble" and "All Too Well (10-Minute Version)."

Beflore playing “All Too Well (10 Minute Version),” Swift talked about bringing all of her favorite memories to France.She did her 2019 “Lover” live show here but hadn’t brought a tour to the country since the Speak Now World Tour 13 years ago.“I cannot thank you enough. I’m so moved," she said.

Follow along with set list changes here .

First set list change

Swift cut "The Archer" from her "Lover" set before beginning her "Fearless" set in a silver and black minidress as opposed to her classic gold one. A graphic showed her "Lover" house with a new addition in the attic for "The Tortured Poets Department." Her "Fearless" set remained the same with "Fearless," "You Belong With Me" and "Love Story."

More: Taylor Swift's 'Love Story' castle will soon host weddings

'Je m'appelle Taylor'

In her monologue before playing the song "Lover," Swift said the night would span 18 years of music. Previously, the concert covered 16 years. Get ready for a "The Tortured Poets Department" addition some time tonight.

She confirmed the tour would still go era by era, and also spoke about how Paramore is a band she idolized since she was a teenager. She ended with some French: "Je m'appelle Taylor."

First new outfit

Swift debuted a new, orange bodysuit and jacket for the "Lover" set.

Snippet of 'Tortured Poets' in intro

A snippet of the song "The Tortured Poets Department" was included in the intro music Swift kicks the concert off with. Is she foreshadowing other changes ahead? It seems likely.

Taylor Swift takes the stage

Swift took the stage about 8 p.m. in Paris (2 p.m. ET) to relaunch her Eras Tour in Europe. She kicked it off with a countdown clock and the Lesley Gore song "You Don't Own Me" as usual. Then she began her "Lover" set with "Miss Americana."

Paramore covers Blondie, Talking Heads in Paris set list

At the Paris show Thursday, Paramore energized fans with eight tracks including their cover off the upcoming “Stop Making Sense” tribute album out May 17 featuring artists like Miley Cyrus and Lorde.

Debuting a blond mullet, lead singer Hayley Williams encouraged fans to dance without fear of judgment ahead of Swift’s performance.

“Enjoy your time with Miss Taylor, we will see you next time,” Williams said. 

Paramore's set list included:

• "Hard Times"
• "Burning Down the House" (Talking Heads cover) 
• "Caught in the Middle"
• “Brick by Boring Brick”
• “The Only Exception”
• “That’s What You Get”
• “Ain't It Fun”
• “This Is Why”

Paramore begins their set

At about 6:45 p.m. in Paris (12:45 p.m. ET), Paramore began their set, officially restarting the second year of the historic Eras Tour. Lead singer Hayley Williams sported a new, blond mullet hairdo.

Paramore opens European leg of Eras Tour

The American rock band is joining Swift on her European leg of 51 shows. Paramore also opened the show for two nights in Glendale, Arizona, in 2023. Their setlist then included nine songs:

• "This Is Why"
• "That’s What You Get"
• "The Only Exception"
• "Still Into You"
• "Rose-Colored Boy"
• "Misery Business"
• "Ain’t It Fun"

Where to watch the Eras Tour livestreams

The Eras Tour is one of the first tours that has been followed, globally, by thousands of fans on digital platforms. Fans dedicate part of their concert experience to livestreaming as Swift performs. Livestream Queen Tess Bohne has spearheaded the "Great War" to find fans going live on social media and broadcasting a custom show to tens of thousands of fans almost every night on her account @tessdear .

On YouTube, user @TaylorSwiftHockeyBro also livestreams the 3-hour show, and on Twitch, @folkleric is a host fans follow.

There is also an app, Swift Alert , that notifies fans who only want to watch certain eras like the acoustic, surprise set. The app also sends out push alerts if there are any major announcements.

What secret songs are left?

For the past 83 shows, Swift has had an acoustic set she customizes every stop with a guitar and piano performance. In 2024, the Eras Tour star added mash-ups of her songs for each stop. Here are the songs she has yet to perform:

• "Taylor Swift" (debut album): "Mary's Song" and "A Perfectly Good Heart"
• "Fearless": "The Way I Loved You," "Change," "Superstar," "We Were Happy," "That's When," "Don't You" and "Bye Bye Baby"
• "Red": "Girl At Home," "Ronan," "Forever Winter" and "Run"
• "1989": "Sweeter Than Fiction"
• "Reputation": "I Did Something Bad"
• "Lover": "London Boy," "Soon You'll Get Better" and "It's Nice to Have a Friend"
• "Folklore": "Hoax"
• "Evermore": "Happiness" and "Closure"
• "Midnights": "Paris" and "Glitch"
• "The Tortured Poets Department": all 31 tracks

Bienvenue dans le 'Eras Tour'

Welcome to the Eras Tour (France's Version). Swift is kicking off her 3-plus-hour concert from La Défense Arena in the Jardins de l’Arche district. The venue is Europe’s largest indoor arena and houses the Racing 92 rugby club. More than 40,000 fans will watch the concert, according to the stadium's website.

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