Let each of

**x**_{1}, x_{2}, x_{3}, …, x_{777}, y_{1}, y_{2}, y_{3}, …, y_{777} be an arbitrary non-zero integer number.

Consider the product

P = (2x_{1}^{2} +3y_{1}^{2}) *
(2x_{2}^{2} +3y_{2}^{2}) * (2x_{3}^{2} +3y_{3}^{2}) * ...* (2x_{777}^{2} +3y_{777}^{2}).

Prove: P cannot be a square number.

(In reply to

Poossible solution by broll)

Very interesting and very original approach!

However: you say "the sequence 'Cheese' contains none"...

How do we know that? (i.e. please substantiate!).

Another approach:

consider a typical expression within brackets modulo 3 and then modulo 2,- show that it is impossible for both the powers of 2 and of 3 within the final product be even while the number of terms is odd.