congruence | Problems in Mathematics

03/18/2017

Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies $2^{n+1}|p-1$.

Problem 344

Let $a, b$ be relatively prime integers and let $p$ be a prime number.
Suppose that we have
\[a^{2^n}+b^{2^n}\equiv 0 \pmod{p}\] for some positive integer $n$.

Then prove that $2^{n+1}$ divides $p-1$.

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Group Theory

12/15/2016

Use Lagrange’s Theorem to Prove Fermat’s Little Theorem

Problem 219

Use Lagrange’s Theorem in the multiplicative group $(\Zmod{p})^{\times}$ to prove Fermat’s Little Theorem: if $p$ is a prime number then $a^p \equiv a \pmod p$ for all $a \in \Z$.

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