A Number Problem

A THEOREM ON POWER SUMS 161 We summarize these results in the following. Theorem. The solutions of (2) are as follows. If p = q, f(x) is a r b i-trary and g(x) = f(x). If p ^ q, the only monic solutions occur when p = 2 and q = 1, in which case f(x) and g(x) are defined by (12), where a is an arbitrary real constant Non-monic solutions for that case can be found using (13). As an example of these results suppose that p = 3 and q = 4. By (14) and (17) we have 13 (n , 4 (3x 2-3x + 1) J , (n = 1, 2, 3, • • •) x=l 1 \ x=l ' \ (4X 3-6x 2 + 4x-1) There are infinite many numbers with the property: if units digit of a positive integer, M, is 6 and this is taken from its place and put on the left of the remaining digits of M, then a new integer, N, will be formed, such that N = 6M. The smallest M for which this is possible is a number with 58 digits (1016949 • • • 677966). 1-4x-x 2 n=o with x = 0,1 we have 1,01016949 * • • 677966, where the period number (behind the first zero) is M.*