Math Is Fun Forum

Primenumbers

Member

Registered: 2013-01-22

Posts: 149

Twin Prime Proof

To generate prime numbers;

A= All primes apart from 2 multiplied together up to prime x.

prime when <(the prime just above x) squared and >1

This is because A is factorable by all possible factors apart from 2, and

isn't. Also

is factorable by 2 and A isn't. The remainders from

for primes up to x carry to the answer. A is multiplied by all odd no.'s so is odd and

is even, generating an odd no. with no factors <itself other than 1.

This concept is useful when trying to prove that there are an infinite number of twin primes;

A= All primes apart from 2 multiplied together up to prime x.
p= prime>x
r=p+1
m= any odd integer

is not factorable by any primes <p. This is because, according to Fermat's Little Theorem, we know that

is factorable by a prime when z+1=that prime. Also we know that remainders for

repeat themselves.
I.e. remainder for 7 for

=

=1

=2

=4

=1

=2

=4

=1............

So when z is a prime it is not a composite so the remainder 1 for other primes <z has not repeated itself. So

will not be factorable by primes <p and

will not be factorable by primes<p but will be factorable by 2.

Therefore

and

generates twin primes, just so long as they = a no. <(the prime just above x) squared and are >1.

Example:

A=105
p=13
r=14
m=157

 

Twin primes because both = <

and >1.

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"Time not important. Only life important." - The Fifth Element 1997

googol

Member

From: Delft, The Netherlands

Registered: 2016-04-22

Posts: 13

Re: Twin Prime Proof

Cool very
The product of all prime numbers up to a certain prime is called a Primorial
In your post

Sometimes the algorithm also works when the result is larger than the next prime squared.

For example:

, p=23, r=22, m=1

and

are both prime but much larger than 23 squared (529).

I am not sure this proves that there is an infinite number of twin primes. The algorithm often finds the same twin primes between 1 and the next largest prime squared. I think you still have to proof that a greater primorial also results in larger twin primes.

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Primenumbers

Member

Registered: 2013-01-22

Posts: 149

Re: Twin Prime Proof

Thanks googol!

Welcome to the forum!

You're right I don't think it proves anything yet.

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"Time not important. Only life important." - The Fifth Element 1997