Select integer x and triangular number y such that 8y=3x^42x^21.
Prove that y is divisible by 28  or find a counterexample.
Since y is triangular let y = t(t+1)/2
8t(t+1)/2 = 3x^42x^21
t^2 + t  (3x^42x^21)/4 = 0
quadratic formula
t = (1 ±
√(1 + 3x^42x^21))/2
t = (1 ± x√(3x^2 2))/2
What is then sought is to find integers for x that make t an integer, so we need 3x^2  2 to be a perfect square
http://oeis.org/A001835 has the formula
a(n) = 4*a(n1)  a(n2), with a(0)=1, a(1)=1
Terms are the solutions to: 3x^22 is a squareIf you take these numbers and put them into √(3x^2 2) you get x
http://oeis.org/A001834 has the formula
b(0) = 1, b(1) = 5, b(n) = 4b(n1)  b(n2)
(I changed a to b to tell them apart.)
So what is sought is to prove t is an integer now that
t = (1 ± a(n)b(n))/2
By inspection a(n) is always odd and b(n) is always odd so this is clearly true.

Posted by Jer
on 20130301 09:22:59 