## Conjecture on when consecutive primes are twin primes

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• RE twin primes conjecture: A property of twin primes, only? Posted by: Jose Ramón Brox ambroxius@terra.es ambroxius Date: Wed Jul 5, 2006 4:51 am (PDT) ...
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RE twin primes conjecture: A property of twin primes, only?
Posted by: "Jose Ramón Brox" ambroxius@... ambroxius
Date: Wed Jul 5, 2006 4:51 am (PDT)

> Let p q consecutive prime numbers p<q.
> Let z=sqrt[(p^2+q^2)/2-1]
>

> Conjecture: p&q ares twin primes IF AND ONLY IF z is
> integer.

No, it's not! You are forgetting the first line of the conjecture. I will
restate it:

KNOWING that p and q are consecutive prime numbers with p < q, then p and q
are twins iff
z=sqrt(p^2+q^2)/2-1) is integer.

Your counterexample isn't valid, since it doesn't fulfill the first
condition: 23 and 25
they aren't consecutive primes!

You can look at it in another way: you select a prime number, and scan for
the next one;
if you compute z and it's an integer, can you conclude that q-p = 2?

Regards. Jose Brox

*************

Thanks for the clarification.

I attempted to analyse this.

Consecutive primes would be of form

p = 6 D -1 and q = 6 D + 6 m + 1

or

p = 6 D -1 and q = 6 D + 6 m + 5.

Perhaps it may be possible to prove that in both cases that if m > 0, that

z = sqrt( [ p^2 + q^2]/2 -1 ) is not rational

and

also to prove that

p = 6 D - 1 and q = 6 D + 5

yields a non-rational z.

My suggestion consists of trying to prove

not that z is a non-integer,

but that z is non-rational.

If the proof falls through, the attempted proof might suggest how to locate
the

counter examples.

Kermit < kermit@... >
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