### 21st Putnam 1960

**Problem B4**

Show that if an (infinite) arithmetic progression of positive integers contains an nth power, then it contains infinitely many nth powers.

**Solution**

Let the difference between adjacent terms of the progression be d. Suppose that N^{n} is an nth power in the progression. Then (N+d)^{n} is a larger nth power in the progression (it is obvious from the binomial expansion that it has the form N + kd).

21st Putnam 1960

© John Scholes

jscholes@kalva.demon.co.uk

25 Jan 2002