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(3): Poossible solution...very creative!
by Ady TZIDON)
I thought someone else might puzzle this out, but apparently not.
The crucial step is to assume a.b.c.d. such that a^2+6b^2 =2c^2+3d^2;
Then 6b^2-3d^2 =2c^2-a^2 and so 3(2b^2-d^2) =2c^2-a^2;
but it is clear that there are no such numbers; (a^2-2b^2)(c^2-2d^2) = (2bd+ac)^2 - 2(ad+bc)^2; the product of two numbers of form (x^2-2y^2) is another number of the same form.
But it is easily computed that 3 is not of that form (though 9 is).
So the two sequences of numbers are distinct. Hence if we find the squares in a^2+6b^2, as already explained, we can be sure that there are no squares in 2c^2+3d^2.
Edited on March 4, 2017, 8:42 am
Posted by broll
on 2017-03-04 08:40:33