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Sum of squares of digits of k in base a representation

Source: China TST 2000, problem 3

May 22, 2005
number theory unsolvednumber theory

Problem Statement

For positive integer a≥2a \geq 2, denote NaN_a as the number of positive integer kk with the following property: the sum of squares of digits of kk in base a representation equals kk. Prove that: a.) NaN_a is odd; b.) For every positive integer MM, there exist a positive integer a≥2a \geq 2 such that Na≥MN_a \geq M.
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