MathDB

For every positive integer

n

b(n)

denote the number of positive divisors of

n

and

p(n)

denote the sum of all positive divisors of

n

. For example, b(14)\equal{}4 and p(14)\equal{}24. Let

k

be a positive integer greater than

1

. (a) Prove that there are infinitely many positive integers

n

which satisfy b(n)\equal{}k^2\minus{}k\plus{}1. (b) Prove that there are finitely many positive integers

n

which satisfy p(n)\equal{}k^2\minus{}k\plus{}1.

Problem Statement