Multiplicative group of a prime field is generated by all primes less than square root plus one

Statement

Suppose $\text{[math]}$ is an odd prime. The multiplicative group modulo $\text{[math]}$ is generated by the congruence classes of all primes less than $\text{[math]}$ .

Related facts and conjectures

Smallest quadratic nonresidue is less than square root plus one
Extended Riemann hypothesis: A stronger form of the Riemann hypothesis that is equivalent to the statement that the smallest quadratic nonresidue modulo $\text{[math]}$ , for any odd prime $\text{[math]}$ , is less than $\text{[math]}$ , where $\text{[math]}$ here is the natural logarithm.
Artin's conjecture on primitive roots: This is a closely related conjecture that states that every integer that is not $\text{[math]}$ or a perfect square occurs as a primitive root for infinitely many primes.

Proof

Given: An odd prime $\text{[math]}$ , $\text{[math]}$ is the subgroup of the multiplicative group modulo $\text{[math]}$ generated by all primes less than $\text{[math]}$ .

To prove: $\text{[math]}$ is the whole multiplicative group modulo $\text{[math]}$ .

Proof: Note first that any positive integer less than $\text{[math]}$ is a product of primes less than $\text{[math]}$ . Hence, its congruence class is in $\text{[math]}$ .

Let $\text{[math]}$ be the smallest positive integer whose congruence class is not in $\text{[math]}$ . Then, $\text{[math]}$ .

Let $\text{[math]}$ be the smallest positive integer such that $\text{[math]}$ , and $\text{[math]}$ . Since $\text{[math]}$ is prime, $\text{[math]}$ , so $\text{[math]}$ . Further, we have:

$\text{[math]}$ .

Since $\text{[math]}$ is not in $\text{[math]}$ , either $\text{[math]}$ or $\text{[math]}$ . However, since $\text{[math]}$ , $\text{[math]}$ , while $\text{[math]}$ . Thus, we have found a positive integer smaller than $\text{[math]}$ that is also not in $\text{[math]}$ , contradicting minimality of $\text{[math]}$ .

Multiplicative group of a prime field is generated by all primes less than square root plus one

Statement

Related facts and conjectures

Proof

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