Overview

Advanced number theory uses structure in modular arithmetic and prime powers. Orders and lifting techniques often simplify exponent problems.

Key Ideas

• If $a^k\equiv1\pmod n$, the least such $k$ is the order of $a$ mod $n$.
• LTE (lifting the exponent) helps with $v_p(a^n-b^n)$ when $p$ divides $a-b$.
• Factorization and parity arguments are constant companions.

Worked Example

Find the largest $k$ with $2^k \mid 3^{12}-1$.

Because $3^2-1=8$, LTE gives $v_2(3^{12}-1)=v_2(3^2-1)+v_2(12)=3+2=5$. So $k=5$.

Practice Problems