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Wilson's Theorem

    This theorem is named after one of Edward Waring's students, John Wilson. But actually, Wilson only observed the result to be true and did not provide its proof. Later Euler proved Wilson's theorem and then Gauss gave a generalization. Basically, the theorem states that the factorial of a prime number minus one is congruent to minus one modulo the prime. Interestingly, its converse is also well-known and gives a sufficient condition for an integer to be a prime.  

Proposition (Wilson's Theorem) If wilson theorem _gr_1.gif] is prime, then wilson theorem _gr_2.gif]

    Proof. The cases wilson theorem _gr_3.gif] and wilson theorem _gr_4.gif] are evident. Choose any integer wilson theorem _gr_5.gif] from the list wilson theorem _gr_6.gif] Recall that the linear congruence wilson theorem _gr_7.gif] has a unique solution wilson theorem _gr_8.gif] since wilson theorem _gr_9.gif] Notice that if wilson theorem _gr_10.gif] and wilson theorem _gr_11.gif] then wilson theorem _gr_12.gif] and so there are exactly wilson theorem _gr_13.gif] pairs left. If we omit the numbers wilson theorem _gr_14.gif] and wilson theorem _gr_15.gif] the effect is to group the remaining integers wilson theorem _gr_16.gif] into pairs wilson theorem _gr_17.gif] and wilson theorem _gr_18.gif] where wilson theorem _gr_19.gif] such that their product wilson theorem _gr_20.gif] When these wilson theorem _gr_21.gif] congruences are multiplied together and the factors are rearranged, we get
    
wilson theorem _gr_22.gif]

and thus wilson theorem _gr_23.gif] whence wilson theorem _gr_24.gif] as desired. wilson theorem _gr_25.gif]  

Example (Wilson's Theorem) Illustrate the proof of Wilson's theorem with wilson theorem _gr_26.gif]

    Solution. For wilson theorem _gr_27.gif] we have wilson theorem _gr_28.gif] which we can prove by using the wilson theorem _gr_29.gif] congruences

wilson theorem _gr_30.gif]

wilson theorem _gr_31.gif]

wilson theorem _gr_32.gif]

wilson theorem _gr_33.gif]

wilson theorem _gr_34.gif]

When these wilson theorem _gr_35.gif] congruences are multiplied together and the factors are rearranged, we get
    
wilson theorem _gr_36.gif]

and thus wilson theorem _gr_37.gif] whence wilson theorem _gr_38.gif] as desired. wilson theorem _gr_39.gif]  

Proposition (Converse of Wilson's Theorem) If wilson theorem _gr_40.gif] is a positive integer such that wilson theorem _gr_41.gif] then wilson theorem _gr_42.gif] is prime.

    Proof. Assume that wilson theorem _gr_43.gif] is not prime; and so there is some nontrivial factor wilson theorem _gr_44.gif] of wilson theorem _gr_45.gif] for which   wilson theorem _gr_46.gif]; whence wilson theorem _gr_47.gif] since wilson theorem _gr_48.gif] for some integer wilson theorem _gr_49.gif] Therefore, wilson theorem _gr_50.gif] can not exist and so wilson theorem _gr_51.gif] is prime. wilson theorem _gr_52.gif]

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Wilson Theorem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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