Sum of how many squares of successive
positive integers is a square number?
Surprisingly enough, trivial answer discounted, only one number qualifies!
I assume the succession of positive integers starts with 1. Then 4900 = 70^2 = 1^2 + 2^2 + ... + 24^2 is the unique nontrivial answer. This makes sense when the second sentence is included.
The classic title of this problem is the Cannonball Problem, and its proof is not easy. The simpler proofs make use of elliptic curves. Specifically the equation suggested by the problem x^2 = y*(y+1)*(2y+1)/6 can be converted to the elliptic curve X^2 = Y^3 - 36Y. 'Elementary' proofs not involving higher order math such as elliptic curves are several pages long and quite involved.
Edited on April 8, 2017, 11:29 pm