Select integer x and triangular number y such that 8y=3x^42x^21.
Prove that y is divisible by 28  or find a counterexample.
(In reply to
SOLUTION +some addons by Ady TZIDON)
My esteemed Ady,
BTW: all the triangle numbers related to this problem are 4/ 3 of a valid triangle number.
Naturally, since that is implicit in the original problem. Assuming the workings from my solution, to save space:
Let 3N+1 = a^2, and let N=n^2n
Let a=(2p1)
3n^23n+1 = 4p^24p+1
Cancelling 1 from both sides:
4p^24p=3n^23n [2]; 4 times a triangular number is 3 times another triangular number.
Checking: a={1,13,181,2521,...}, b={1,15,209,2911,...}, n={1,8,105,1456,...}, p=1,7,91,1261,...}, x={1,3,11,41,..}, y={0,56,10920,2118480,...}. From which we can see at once, for example, that 4*91*90=3*105*104, etc., etc. Hence the statements 8y=3x^42x^21 and 4p^24p=3n^23n are essentially equivalent.

Posted by broll
on 20130302 06:22:23 