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 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. 

Posted by Ady TZIDON on 2017-02-27 04:41:25