Wilson's Theorem, and introducing another standard term: the Wilson quotient,  

Wilson's (the discoverer, not the prover) theorem (around 1770). Every prime satisfies (mod ). 

Remark. It was something of a surprise for me to hear you had not encountered this in your second year NT course. 

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L := []: for p from 2 to 50 do if isprime(p) then             R||p := PRFAC(1, p-1, p, 1):           ### COMPUTES THE RESIDUE OF THE FACTORIAL                 L := [op(L), p]: fi od: print(``); print(array([ ['p', ``, ``, '(p-1)!', ``, ``, 'residue'], seq([p, ``, ``, (p-1)!, ``, ``,R||p], p = L)])): print(``); lprint(`The final column show the LEAST ABSOLUTE residue of (p-1)! mod p.`);

 

 

 

 

`The final column show the LEAST ABSOLUTE residue of (p-1)! mod p.`

 

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A standard term. Following from the above, simply translating congruence into its equation equivalent, we have: 

for some integer , giving: 

That integer , is known as the Wilson quotient  : 

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

 

 

 

	(5.3.1)

 

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