RE twin primes conjecture: A property of twin primes, only?
Posted by: "Jose Ramón Brox" 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
No, it's not! You are forgetting the first line of the conjecture. I will
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
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
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
Kermit < kermit@...