a square form

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Fm Jakaria
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a square form

Unread post by Fm Jakaria » Fri Nov 14, 2014 10:49 pm

Determine all perfect squares of form $4xy-(x+y)$ with $x,y$ positive integers.
You cannot say if I fail to recite-
the umpteenth digit of PI,
Whether I'll live - or
whether I may, drown in tub and die.

User avatar
Fm Jakaria
Posts:79
Joined:Thu Feb 28, 2013 11:49 pm

Re: a square form

Unread post by Fm Jakaria » Sun Nov 16, 2014 7:43 pm

Let $4xy - (x+y) = z^2$ with z nonnegative.
Then $x = \dfrac {y+z^2} {4y-1}$ is a positive integer. Here $4y-1$ is an odd positive integer >1; with not all prime divisor of form 4k+1 for positive integer k; else 4y-1 would itself have this form. Let p be a prime divisor of 4y-1 of form 4k-1 for positive integer k. Then $p|4y-1|y+z^2$.
Now $modp, z^2 \equiv -y \equiv -\dfrac{1}{4}$.So $(2z)^2\equiv -1$.The last is impossible as -1 is nonzero mod p and quadratic nonresidue mod p. :mrgreen:
You cannot say if I fail to recite-
the umpteenth digit of PI,
Whether I'll live - or
whether I may, drown in tub and die.

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