### 14th Putnam 1954

**Problem B1**

Show that for any positive integer r, we can find integers m, n such that m^{2} - n^{2} = r^{3}.

**Solution**

*Easy.*

We notice that m^{2} - n^{2} = (m + n)(m - n). This suggests taking m + n = r^{2}, m - n = r. This works: m = r(r + 1)/2, n = r(r - 1)/2.

14th Putnam 1954

© John Scholes

jscholes@kalva.demon.co.uk

24 Nov 1999