Prove that there cannot exist any positive integer x, such that each of 2x² + 1, 3x² + 1 and 6x² + 1 is a perfect square.

In 1768, Lagrange gave the first complete proof of the solvability of
x² - Dy² = 1  ( complicated)

I erased a wrong statement - please see my next post

OUT OF THE  THREE GIVEN EQUATIONS WE CAN FULLFIL THE FIRST  AND THE LAST e.g. D=2 y=2 =>>x^2=1+8=9   OK
D=6 y=2  ==>>x^2=1+24=25               OK
but not the second x^2=1+12=13   NOT A SQUARE

Rem: puzzle's x is our y ; D=2,3,6

Edited on January 10, 2010, 1:13 pm

Posted by Ady TZIDON on 2010-01-03 19:41:01