iff all coefficients of the difference polynomial are divisible by .
Remark 1. The formal congruence of polynomials is an equivalence relation in the set
Remark 2. The formal congruence (1) implies that all integers substituted for the variables satisfy it, in other words, one can speak of an identical congruence. However, there are identical congruences that are not formal congruences; e.g.
where is a positive prime.
If one substitutes in this formal congruence 1 for all , , …, , one obtains the congruence
substitution of shows that the last congruence is valid also for negative integers . If it is supposed that is not factor of , we have got the Fermat’s little theorem.
Let be a positive prime number. It is possible that the degree congruence
where and , has modulo incongruent roots . We then have the formal congruence
Especially, we have
because Fermat’s little theorem guarantees the roots for the congruence . If the value is substituted in the previous formal congruence, it gives
which is half of Wilson’s theorem. The reverse part of this theorem follows from the last congruence so that if were not prime, then the number would be divisible by any proper divisor of .
- 1 F. Stöwener: “Simultanbeweis des Fermatschen und Wilsonschen Satzes”. – Elemente der Mathematik 30 (1975).
|Date of creation||2013-03-22 14:23:50|
|Last modified on||2013-03-22 14:23:50|
|Last modified by||pahio (2872)|