Talk:Prime number

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I struck my own comment as it was useless (due to simple sleight of hand of ref) —Preceding unsigned comment added by Billymac00 (talk • contribs)

Successive composite numbers

Ranjitr303 (talk) 07:07, 24 June 2010 (UTC)can somebody tell me a formula for finding n succesive numbers such that within the n succesive numbers none is a prime. i had read about this in some book but i am not sure whether it is related to the great Ramanujan ?

See Prime number#Gaps between primes. For any positive integer m ≥ 2, none of the m − 1 numbers from m! + 2 to m! + m can be prime, because m! + 2 is divisible by 2, m! + 3 is divisible by 3, and so on. So to find a sequence of n consecutive composite numbers, just take m ≥ n + 1. For example, the 10 numbers from 11! + 2 = 39916802 to 11! + 10 = 39916811 are all composite numbers. Gandalf61 (talk) 08:03, 24 June 2010 (UTC)

Irrationality of π implies infinite number of primes?

It is claimed that there is an equation

${\frac {\pi }{4}}\ =({\frac {1}{1+{\frac {1}{3}}}})({\frac {1}{1-{\frac {1}{5}}}})({\frac {1}{1+{\frac {1}{7}}}})({\frac {1}{1+{\frac {1}{11}}}})\cdots \;$

(- when primes of the form 4k+1,and + when primes of the form 4k+3)

which leads to the irrationality of π implying the infinitude of primes. That equation does not seem to be sourced; even if it were, the equation:

$\zeta (2)={\frac {\pi ^{2}}{6}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-2}}}$

provides a much simpler proof from the transcendence of π to the infinitude of primes. — Arthur Rubin (talk) 18:21, 14 September 2010 (UTC)

If we have already derived the Euler product for ζ, then there are much simpler ways of inferring infinity of primes than using the (fairly nontrivial) transcendence of π.—Emil J. 18:57, 14 September 2010 (UTC)

I think the given equation has been derived by applying the Euler product to Gregory's series

${\frac {\pi }{4}}=\sum _{k=0}^{\infty }{\frac {1}{4k+1}}-\sum _{k=0}^{\infty }{\frac {1}{4k+3}}$

noting that each odd integer is the product of odd primes, and an integer of the form 4k+3 must have an odd number of prime factors of the form 4k+3 (counting repeated factors with appropriate multiplicity). But I agree that this claim definitely needs a reliable source, and if we are using Euler products then the non-convergence of the harmonic series is a much more direct demonstration of the infinitude of primes. Gandalf61 (talk) 09:19, 15 September 2010 (UTC)

If anything like this is to be in I'd have thought the best example would be the proof from that log x has no bound as x goes to infinity that is in 'Proofs from the Book'. I think one could make a case that proofs in that are notable. A proof from that pi is not rational is just silly considering how much work one needs to prove the antecedent. Dmcq (talk) 12:40, 15 September 2010 (UTC)

By definition?

I never understood why the number 1 is not a prime number.

I always get "The number 1 is by definition not a prime number. "

But why? Why did they decide to not make 1 a prime number. It would seem so natural and complete to allow the number 1 to be a prime number. —Preceding unsigned comment added by 208.251.83.66 (talk) 23:43, 27 September 2010 (UTC)

Making 1 prime breaks uniqueness of prime factorization. See Prime number#Primality of one.—Emil J. 10:26, 28 September 2010 (UTC)

What? 6 isn't prime number thne?

Six is divisible by six and one, why isn't it a prime number? 78.110.160.85 (talk) 21:26, 29 November 2010 (UTC)