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I ran into an interesting little math problem yesterday: Find all integers N such that 1+2+…+N is the square of another integer.

The summation is simple, and anyone who received a little math education can immediately convert it into N(N+1)/2. Now the problem becomes finding N such that

$latex \frac{N(N+1)}{2} = M^2$

where M is another integer. I spent 2 hours yesterday in the afternoon trying to figure out the general solution with a friend. In the end we were only able to find half the solution. Still it might be useful to record what we found here.

First is notice that N and N+1 are relatively prime. One of them has to be even, and after dividing by 2 it is still relatively prime with the other. Therefore the two numbers have to be squares of a different integer. This splits the problem into two different scenarios:

First: $latex…

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