MathDB

Classical NT in XMO

Source: XMO2024

April 30, 2024

number theory

Problem Statement

Let

p

be a prime number and

k

is a integer with

p|2^k-1

for

a\in \{ 1,2,\ldots,p-1\}

, let

m_{a}

be the only element that satisfies

p|am_{a}-1

and define

T_{a}= \{ x \in \{1,2,\ldots p-1\} | \{ \frac {m_{a}x}{p} - \frac {x}{ap} \} < \frac{1}{2}

and there exists integer y satisfying p | x-y^

k+1

\}

Try to proof that there exists an integer

m

and integers

1 \le a_1 <a_2< \ldots < a_{m} \le p-1

satisfying

|T_{a_1}| = |T_{a_2}| = \ldots = |T_{a_{m}}| = m

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