This is an English translation of Euler's ``Theoremata circa residua ex
divisione potestatum relicta'', Novi Commentarii academiae scientiarum
Petropolitanae 7 (1761), 49-82. E262 in the Enestrom index.
Euler gives many elementary results on power residues modulo a prime number
He shows that the order of a subgroup generated by an element a in F\_p^* must
divide the order p-1 of F\_p^* (i.e. a special case of Lagrange's theorem for
Euler also gives a proof of Fermat's little theorem, that a^p-1 = 1 mod p
for a relatively prime to p (i.e. not 0 mod p). He remarks that this proof is
more natural, as it uses multiplicative properties of F\_p^* instead of the
Thanks to Jean-Marie Bois for pointing out some typos.