--- In

primenumbers@yahoogroups.com, Sebastian Martin Ruiz <s_m_ruiz@...> wrote:

>

> ________________________________

>

> Prove:

>

> Theorem:

>

> Let c a positive even number >2

>

> If (x,y)= (1,1) and (c^2/2-1,3c^2/2-2c-1) are the only positive integer solutions

>

> of the polynomial

>

> -3x^2+y^2-2xy-4cx+4cy+4=0

>

> then c +1 and c-1 are twin primes

>

>

> Sincerely

>

> Sebastián Martín Ruiz

>

Based on Warren's (corrected) simplification to your equation

-3x^2 + y^2 -2xy - 4cx + 4cy + 4 = 0

namely

r^2 - x^2 - c^2 + 1 = 0

Then

(r-x)(r+x) = (c-1)(c+1)

If both c-1 and c+1 are prime, then clearly there are only two options:

r-x = c-1 and r+x = c+1

OR

r-x = 1 and r+x = (c-1)(c+1)

Follow each of those options and you easily arrive at the two results.

(x,y) = (1,1) and ( (c^2)/2 - 1,3(c^2)/2 - 2c - 1)

Mark