MathDB

Innovative or not so much NT with powers and remainders

Source: RMM 2024 Problem 4

February 29, 2024

RMMnumber theoryremaindersize

Problem Statement

Fix integers

a

and

b

greater than

1

. For any positive integer

n

, let

r_n

be the (non-negative) remainder that

b^n

leaves upon division by

a^n

. Assume there exists a positive integer

N

such that

r_n < \frac{2^n}{n}

for all integers

n\geq N

. Prove that

a

divides

b

.
Pouria Mahmoudkhan Shirazi, Iran