Olympiad problems

PEN B Problems, 3

Show that for each odd prime $p$, there is an integer $g$ such that $1<g<p$ and $g$ is a primitive root modulo $p^n$ for every positive integer $n$.

PEN B Problems, 7

Tags: algebra , polynomial , modular arithmetic , Primitive Roots

Suppose that $p>3$ is prime. Prove that the products of the primitive roots of $p$ between $1$ and $p-1$ is congruent to $1$ modulo $p$.

2023 Indonesia TST, N

Tags: number theory , Fermat s Little Theorem , primitive root , Primitive Roots

Let $p,q,r$ be primes such that for all positive integer $n$, $$n^{pqr}\equiv n (\mod{pqr})$$ Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$

2015 IFYM, Sozopol, 5

Tags: number theory , Primitive Roots

Let $p>3$ be a prime number. Prove that the product of all primitive roots between 1 and $p-1$ is congruent 1 modulo $p$.

2023 Indonesia TST, N

Tags: number theory , Fermat s Little Theorem , primitive root , Primitive Roots

Let $p,q,r$ be primes such that for all positive integer $n$, $$n^{pqr}\equiv n (\mod{pqr})$$ Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$

PEN B Problems, 4

Tags: modular arithmetic , Primitive Roots

Let $g$ be a Fibonacci primitive root $\pmod{p}$. i.e. $g$ is a primitive root $\pmod{p}$ satisfying $g^2 \equiv g+1\; \pmod{p}$. Prove that [list=a] [*] $g-1$ is also a primitive root $\pmod{p}$. [*] if $p=4k+3$ then $(g-1)^{2k+3} \equiv g-2 \pmod{p}$, and deduce that $g-2$ is also a primitive root $\pmod{p}$. [/list]

PEN B Problems, 6

Tags: modular arithmetic , number theory , relatively prime , Primitive Roots

Suppose that $m$ does not have a primitive root. Show that \[a^{ \frac{\phi(m)}{2}}\equiv 1 \; \pmod{m}\] for every $a$ relatively prime $m$.

PEN B Problems, 1

Tags: quadratics , modular arithmetic , Primitive Roots

Let $n$ be a positive integer. Show that there are infinitely many primes $p$ such that the smallest positive primitive root of $p$ is greater than $n$.