Denote by the greatest prime number not exceeding . By Bertrand’s postulate there is a prime with . Therefore we have . If were an integer, then the sum
had to be divisible by . However its addend is not divisible by but all other addends are, whence the sum cannot be divisible by . The contradictory situation means that is not integer when .
is always divisible by .
Proof. Consider the polynomial
It may be written by Wilson’s theorem as
being thus true for any integer . From (4) one can successively infer that divides all coefficients , i.e. that (4) actually is a formal congruence.
For the derivative of the polynomial one has
By (1), this equation implies
Since , one has . It then follows by (6) that . And since (5) divided by gives
the assertion has been proved.
- 1 L. Kuipers: “Der Wolstenholmesche Satz”. – Elemente der Mathematik 35 (1980).
|Date of creation||2013-03-22 19:14:06|
|Last modified on||2013-03-22 19:14:06|
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