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July 18th, 2014, 09:38 AM   #1
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Another prime theory

First the facts:

a is element of IN

a^3+(a-1)^3=(2a-1)*(a^2-a+1)

x=a^3+(a-1)^3
y=(2a-1)
z=(a^2-a+1)

x=y*z

Ok, now my new theory:

If (x-1)/3 or (x+1)3 is element of IN,
then x have one primefactor in the form of y or z!

It is similar to:

If the cross sum of x is completely divisible by 9,
then it is the only case were x have none primefactor in the form of y or z!


To see what i mean you can watch the table..

Have fun with prime research^^
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July 18th, 2014, 10:13 AM   #2
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What is IN? Is this the integers $\mathbb{Z}$, the positive integers $\mathbb{Z}^+$, the natural numbers $\mathbb{N}=\{0,1,\ldots\}$, or something else?

It seems that you are saying:

If a^3+(a-1)^3 is not divisible by 3, then either 2a-1 or a^2-a+1 is prime.

Is this correct?
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July 18th, 2014, 12:54 PM   #3
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IN is the natural numbers.

yes or lets say

If a or a+1 is divisible by 3, then either 2a-1 or a^2-a+1 is prime.

But i found to much composites, often when y is divisible by 5, also when y is p^k

Maybe just luck in a small research



But this is interesting:

When log((a^2-a+1)+1)=log(z+1) is element of IN, then log(z+1) is an Mersenne exponent.

There are Mersenne Primes z for a={2;3;6;91;...}

Last edited by PerAA; July 18th, 2014 at 01:16 PM.
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July 19th, 2014, 05:45 PM   #4
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strike the log from both sides, it's irrelevant. Check your claim, I don't think 91 works.
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July 19th, 2014, 05:49 PM   #5
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Quote:
Originally Posted by PerAA View Post
IN is the natural numbers.

yes or lets say

If a or a+1 is divisible by 3, then either 2a-1 or a^2-a+1 is prime.

But i found to much composites, often when y is divisible by 5, also when y is p^k

Maybe just luck in a small research
Indeed. The conjecture fails infinitely often, indeed on a subset of the naturals with density 1.

Quote:
Originally Posted by PerAA View Post
When log((a^2-a+1)+1)=log(z+1) is element of IN, then log(z+1) is an Mersenne exponent.

There are Mersenne Primes z for a={2;3;6;91;...}
Is this for any integers a and z? What is the base of the logarithm?
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