Euler's criterion

For an odd prime

p

and

p ∤ a

a^{(p - 1) /2} \equiv (\frac{a}{p}) (mod p) .

(1)

Table of contents
- Primes congruent to five modulo eight from odd squares plus four Euler's criterion

Primes congruent to five modulo eight from odd squares plus four

 0  0

Every odd prime factor of an odd square plus four is one modulo four. The integer is five modulo eight, so at least one prime factor is five modulo eight. Choosing the odd square divisible by any proposed finite list excludes the list and proves infinitude.