Proofs
Both of the proofs (for prime moduli) make use of the fact that the residue classes modulo a prime number are a field. See the article prime field for more details. Lagrange's theorem (in any field a polynomial of degree n has at most n roots) is needed for both proofs.
Read more about this topic: Wilson's Theorem
Famous quotes containing the word proofs:
“To invent without scruple a new principle to every new phenomenon, instead of adapting it to the old; to overload our hypothesis with a variety of this kind, are certain proofs that none of these principles is the just one, and that we only desire, by a number of falsehoods, to cover our ignorance of the truth.”
—David Hume (1711–1776)
“I do not think that a Physician should be admitted into the College till he could bring proofs of his having cured, in his own person, at least four incurable distempers.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“Would you convey my compliments to the purist who reads your proofs and tell him or her that I write in a sort of broken-down patois which is something like the way a Swiss waiter talks, and that when I split an infinitive, God damn it, I split it so it will stay split, and when I interrupt the velvety smoothness of my more or less literate syntax with a few sudden words of bar- room vernacular, that is done with the eyes wide open and the mind relaxed but attentive.”
—Raymond Chandler (1888–1959)