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Saturday, February 27, 2016

Triangular numbers

A triangular number T(n) is defined as n(n+1)/2.  It is equal to the binomial coefficient (n+12) and is also equal to  0+1+2++n. When is an integer m0 equal to a triangular number T(n) for some integer n0?  We have n(n+1)=2m. Using the quadratic formula to solve the quadratic equation n2+n2m=0 for n, we obtain one solution for n:
n=8m+112
The other solution for n is negative (or complex) so we will not use that.
If 8m+1 is not the square of an integer, then n is not an integer.  If 8m+1 is the square of an integer, then 8m+1 is odd and thus 8m+1 is odd as well. This means that n=8m+112 is an integer. Thus m is a triangular number if and only if 8m+1 is the square of an integer.

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