Given the positive integers, x, y, and z, are consecutive terms of an arithmetic progression, the least value of the positive integer, n,
for which the equation, x^2 – y^2 – z^2 = n
, has exactly two
solutions is n=27
since 34^2 – 27^2 – 20^2 = 27
and 12^2 – 9^2 – 6^2 = 27
It turns out that n=1155
is the least value for which there are exactly ten
How many values of n less than one million have exactly ten distinct solutions?
Source: Project Euler
Set x=y+d and z=y-d and substitute to get y(4d-y)=n. By looking at the factors of n we can readily find solutions, such as (40,21,2) and (482,385,288) for n=1155.
For n with exactly 10 solutions we need exactly 10 values for y. That lets out cases where n=prime or n=product of exactly 2 or 3 primes. Not all factors are solutions either because z=y-d>0, so that 4y>4d, 3y>(4d-y), and 3y^2>n. In the case of n=1155 that means y>19, excluding possible solutions 1,3,5,7,11,15.
I don't see how to get closer to a full count without a computer program.
Posted by xdog
on 2015-08-24 12:15:16