MathDB

Let

m

be a positive integer. a. Show that there exist infinitely many positive integers

k

such that

1+km^3

is a perfect cube and

1+kn^3

is not a perfect cube for all positive integers

n<m

. b. Let

m=p^r

where

p \equiv 2 \pmod 3

is a prime number and

r

is a positive integer. Find all numbers

k

satisfying the condition in part a.

Problem Statement