## A different approach to the twin primes conjecture

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• Here s a rather different approach than Yitang Zhang s. Probably unlike his it is hopeless, but anyhow I will explain the idea. The goal is to fix a small
Message 1 of 2 , Jun 4, 2013
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Here's a rather different approach than Yitang Zhang's.
Probably unlike his it is hopeless, but anyhow I will explain the idea.

The goal is to fix a small constant integer K (optimally K=2, but anything
smaller than, say, 60 will be impressive) then prove that there exist an infinite number of
prime-pairs {P, P+K}. Call these "K-twins." We also with the approach below could
be able to attack constellations involving more than two primes, e.g. "triplets"
and also could attack things like "primes p where 2*p+1 is prime also."

If N=1 mod 4 is squarefree, then N is prime if N has exactly 4 representations
of the form N=4*a^2+b^2 with a,b integers. Also, it is not possible for
any such N, whether prime or not, to have 1,2,3,5,6,7 reps.

If N=1 mod 6 is squarefree, then N is prime if N has exactly 4 representations
of the form N=3*a^2+b^2 with a,b integers. Also, it is not possible for
any such N, whether prime or not, to have 1,2,3,5,6,7 reps.

There are many theorems of this general ilk that arise from unique factorization
theorems in quadratic number fields with class number 1.

Now certain "theta functions" give generating functions for these counts of representations. That is of interest because usually with primes you mess with
zeta functions and Dirichlet L functions, which are horrible. Theta functions
are far nicer in many ways.

For example if theta(x)=1+2x+2x^4+2x^9+2x^16+2x^25+...
then F(x)=theta(x)*theta(x^4) counts representations of N of form N=4*a^2+b^2.

A difficulty in all this is the fact those theorems only work for SQUAREFREE N.
However, if we were to prove theorems not about "primes" but rather about "primes and numbers with square factors", ("primeanwsfs") those would still be interesting theorems,
and you might be able to surmount the difficulty with some kind of sieving argument later (concerning sieving out numbers with square factors).

Now if F(x) is such a generating function and we want to prove there exist an
infinite number of primeanwsfs-pairs (P, P+K),
then one way would be to show an infinite number of powers x^p of x exist in the
series expansion of (1/2+x^K)*F(x), which have coefficient=6.
• Here s a rather different approach than Yitang Zhang s. Probably unlike his it is hopeless, but anyhow I will explain the idea. The goal is to fix a small
Message 2 of 2 , Jun 4, 2013
• 0 Attachment
Here's a rather different approach than Yitang Zhang's.
Probably unlike his it is hopeless, but anyhow I will explain the idea.

The goal is to fix a small constant integer K (optimally K=2, but anything
smaller than, say, 60 will be impressive) then prove that there exist an infinite number of
prime-pairs {P, P+K}. Call these "K-twins." We also with the approach below could
be able to attack constellations involving more than two primes, e.g. "triplets"
and also could attack things like "primes p where 2*p+1 is prime also."

If N=1 mod 4 is squarefree, then N is prime if N has exactly 4 representations
of the form N=4*a^2+b^2 with a,b integers. Also, it is not possible for
any such N, whether prime or not, to have 1,2,3,5,6,7 reps.

If N=1 mod 6 is squarefree, then N is prime if N has exactly 4 representations
of the form N=3*a^2+b^2 with a,b integers. Also, it is not possible for
any such N, whether prime or not, to have 1,2,3,5,6,7 reps.

There are many theorems of this general ilk that arise from unique factorization
theorems in quadratic number fields with class number 1.

Now certain "theta functions" give generating functions for these counts of representations. That is of interest because usually with primes you mess with
zeta functions and Dirichlet L functions, which are horrible. Theta functions
are far nicer in many ways.

For example if theta(x)=1+2x+2x^4+2x^9+2x^16+2x^25+...
then F(x)=theta(x)*theta(x^4) counts representations of N of form N=4*a^2+b^2.

A difficulty in all this is the fact those theorems only work for SQUAREFREE N.
However, if we were to prove theorems not about "primes" but rather about "primes and numbers with square factors", ("primeanwsfs") those would still be interesting theorems,
and you might be able to surmount the difficulty with some kind of sieving argument later (concerning sieving out numbers with square factors).

Now if F(x) is such a generating function and we want to prove there exist an
infinite number of primeanwsfs-pairs (P, P+K),
then one way would be to show an infinite number of powers x^p of x exist in the
series expansion of (1/2+x^K)*F(x), which have coefficient=6.
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