18177Re: [PrimeNumbers] A property of prime twins, only?
- Jul 1, 2006
> Let p, q be consecutive prime numbers, p<q.Since p and q are twin primes q=p+2
> Let z=sqrt[(p^2+q^2)/2-1]
> Conjecture: p&q are prime twins iff z is integer.
So z becomes
sqrt( (p^2 + (p+2)^2)/2 -1) =
sqrt( (p^2 + p^2 + 4p +4)/2 -1) =
sqrt( (2p^2+4p+4)/2 -1 ) =
sqrt(p^2+2p+2 - 1) =
sqrt( (p+1)^2 ) =
Now let p and q be any (non-twin!) consecutive primes with 2 < p < q
Now there exist an n>1 such that q=p+2*n
Doing the same as above leads to:
z = sqrt((p^2 + p^2+4np+ 4n^2 )/2 -1)
Its roots are -n +/- sqrt( (-n)^2 - 2n^2+1 ) = -n +/- sqrt( n^2 - 2n^2+1 )
= -n +/- sqrt( -n^2+1 )
which has solutions in integers only for n=0,1
So the conjecture is indeed true.
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