Let each of x1, x2, x3, …, x777, y1, y2, y3, …, y777
be an arbitrary non-zero integer number.
Consider the product
P = (2x12 +3y12) *
(2x22 +3y22) * (2x32 +3y32) * ...* (2x7772 +3y7772).
Prove: P cannot be a square number.
(In reply to re(2): Poossible solution...very creative!
y cannot be 0 , the text implies non-zero integers.
Still, properly modified, you might converge to formal proof.
Please, while editing erase the 1st 0 in the brackets- correcting the pOOsible word is impossible!
My solution (took me a long time to reach it) is much simpler - just notice that 2x^2 is never 1 mod 3.