Is there any pattern to the distribution of prime numbers?

Question

Is there any pattern to the distribution of prime numbers?

Answer 1

It would appear at this point that the distribution is unpredictable.

In addition, to make things weirder, primes also tend to show up as what are called twin primes. Every so often, both the numbers "n" and "n+2" will be prime. There is a theory, yet to be proven, that there is an infinite number of twin primes known as the "twin prime conjecture" (who said mathematicians were boring?).

To disagree, though, in order to determine whether or not a number is prime, one must divide it by all of the known primes less than the number's square root. This seems non-intuitive, so let's look at an example.

Suppose that we have a suspected prime number, C, which is rather large. In order to prove that C is prime, one must start dividing it by all of the prime numbers. Therefore, one would start at 2, then 3, 5, 7, 11, 13, etc. If any of these divide into C, then "Bingo" it's composite. If they don't, then we need to keep plugging away. So, if we now find a prime, P, that divides into C, then the remainder must be larger than or equal to P. Why? Well, since we've tried every factor less than P, the remaining factor(s) must be at least as big as P.

Ok, so what's the largest that P can be? If the remainder after you've must be at least as big as P, then the smallest the remainder can be is P. This would mean that C is just P^2. Therefore, you would only have to test primes up to sqrt(C). (Note: if sqrt(C) comes out even, then it's a sure bet that it's a composite number).

Answer 2

The products of the first few primes result in numbers with interesting patterns:

7 · 11 · 13 = 1,001
1 · 2 · 3 · 5 · 7 · 11 · 13 = 30,030
1 · 2 · 3 · 5 · 7 · 11 · 13 · 17 = 510,510
1 · 2 · 3 · 5 · 7 · 11 · 13 · 17 · 19 = 9,699,690
(1 · 2 · 3 · 5 · 7 · 11 · 13)^3 = 27,081,081,027,000

Don't know if this is useful, but interesting nonetheless.

Answer 3

that is extra consumer-friendly to describe primes as being of the variety 6n±a million, this receives rid of all unusual multiples of three from being considered as they're all 6n±3 38 is the smallest answer on your question because it produces seventy seven this is 7 x 11 40 5 is yet another because it produces ninety one this is 7 x 13 fifty 9 is yet another because it produces 119 this is 7 x 17 sixty six is yet another because it producxes 133 whicj is 7 x 19 71 is yet another because it produces 143 this is 11 x 13 80 is yet another because it produces 161 this is 7 x 23 ninety 3 is yet another because it produces 187 this is 11 x 17 those are the in straight forward words solutions below one hundred, it appears that evidently there an unlimited variety of solutions, as there are countless variety of primes and countless variety of options of picking 2 or extra primes to multiply them jointly. 500 ensuing in 1001 this is 7 x 11 x 13 is the smallest answer with 3 excellent factors more desirable than 5 8508 ensuing in 17017 this is 7 x 11 x 13 x 17 is the smallest answer with 4 excellent factors more desirable than 5. right here's a suitable qiestion. In any run of ten numbers the optimal variety of primes is 4 as all numbers ending in 5 at the instantaneous are not excellent and all even numbers at the instantaneous are not excellent. subsequently 11 13 17 19 are all excellent yet in 21 23 27 29 in straight forward words 23 and 29 are excellent Is 11 13 17 19 the in straight forward words such run of four successive primes? If no longer, locate others and evaluate no matter if there is an unlimited variety of such runs of four primes or nor?

Answer 4

Well, that depends on what you mean by pattern.

Here are three that may be of interest.

1. 2 is the only even prime. By the definition of even numbers, all other prime numbers must be odd.

2. The Sieve of Eratosthenes' (look that up for yourself) can be used to determine which positive integers are not prime, and therefore, the others are prime.

3. All prime numbers must be positive.

While you might consider that these are trivial, they are 'patterns'.

Answer 5

None that is known. If there is a pattern its discovery would be a huge breakthrough in mathmatics. Many encryption algorithms depend on it being very difficult to determine if a large odd number is prime or not. Pretty much the only way to determine if a number is prime or not is to try to divide it with all of the known prime numbers that are less than half of the number under examination.

Answer 6

The most important result in the distribution of primes is the "prime number theorem" (PNT), which says that the number of primes less than n converges to (n / ln n) as n grows large. You can find much more information about the history and variations of this theorem by Googling "prime number theorem".

Although you didn't ask about primality testing, several answerers suggested (incorrectly) that the Sieve of Eratosthenes (testing primality of n by dividing it by all primes less than n) is the preferred way to test primality. In fact, it is hopelessly inefficient for large numbers (exponential in the number of digits). There are several algorithms that test primality of a k-digit number in time polynomial in k; Google "primality testing" for details.

Answer 7

Solving the "equation" for prime numbers is one of the math world's biggest challenges. Check out the Clay Institute - they have a $1,000,000 prize for solving this one (actually it is the Riemann Hypothesis which has yet to be proven).

As of the last few months, the largest prime number discovered to date has millions of digits!

So the short answer is "we don't know."