Fermat's 'little' Theorem, and introducing a standard term: the Fermat quotient,  

Fermat's (the discoverer, not the prover) little theorem (mid 1600's). For every prime and integer (mod ) satisfies 

(mod ) 

(I understand you will have encountered this in your second year course on Number Theory.) 

In particular, for every odd prime we have 

(mod ) 

A standard term. From the just seen congruence, translating into its equation equivalent, we have: 

for some integer , giving: 

That integer , is known as the Fermat quotient to the base 2, : 

>	L := []: for p from 3 to 50 do if isprime(p) then             q||p := (2^(p-1) - 1)/p:                 L := [op(L), p]: fi od:      print(``); print(array([['p', ``, ``, 'q[p](2)'],           seq([p, ``, ``, q||p], p = L)])): print(``);

 

 

 

	(5.1.1)

 

>

 

A very quick note in passing. Primes for which are called Wieferich primes  

                                                (see, e.g.,  https://en.wikipedia.org/wiki/Wieferich_prime) 

They have a connection with the famous 'Fermat's last Theorem' ... .  

Only two are known, 1093 and 3511 (the current search limit is ): 

>	print(``); p := 1093; print(``);(2^(p-1) - 1)/p; print(``);  (2^(p-1) - 1)/p mod p;

 

 

 

 

 

 

	(5.1.2)

 

>

 

>	print(``); p := 3511; print(``);(2^(p-1) - 1)/p; print(``);  (2^(p-1) - 1)/p mod p;

 

 

 

 

 

 

	(5.1.3)

 

>