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.

Let the sequence  x^2 + 6y^2 {1, 4, 6, 7, 9, 10, 15, 16,..} be 'Chalk'; let the sequence 2x^2 + 3y^2 {0, 2, 3, 5, 8, 11, 12, 14, 18, 20...)be 'Cheese'.

Cheese*Cheese = Chalk; e.g. = 2*2 = 4, 5*2=10.

Cheese*Chalk = Cheese; e.g.2*4=8,4*3=12,2*10=20.

The sequence 'Chalk' contains every square; the sequence 'Cheese' contains none.

Hence P cannot be square.

Posted by broll on 2017-02-27 01:21:33