formal congruence

Let m be a positive integer.  Two polynomialsf(X1,X2,,Xn)  and  g(X1,X2,,Xn)  with integer coefficients are said to be formally congruent modulo m, denoted

f(X1,X2,,Xn)¯g(X1,X2,,Xn)(modm), (1)

iff all coefficients of the difference polynomial  f-g  are divisible by m.

Remark 1.  The formal congruence of polynomials is an equivalence relationMathworldPlanetmath in the set  [X1,X2,,Xn].

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 p is a positive prime.


  1. 1.

    (X+Y)p¯Xp+Yp(modp)  when p is a positive prime numberMathworldPlanetmath.  This result is easily proved with the binomial theoremMathworldPlanetmath (cp. the freshman’s dream) and may be by inductionMathworldPlanetmath generalized to


    If one substitutes in this formal congruence 1 for all X1, X2, …, Xn, one obtains the congruenceMathworldPlanetmathPlanetmathPlanetmath


    substitution of -1 shows that the last congruence is valid also for negative integers n.  If it is supposed that p is not factor of n, we have got the Fermat’s little theoremMathworldPlanetmath.

  2. 2.

    Let p be a positive prime number.  It is possible that the nth degree congruence

    P(x):=a0xn+a1xn-1++an 0(modp),

    where  aii  and  pa0,  has n modulo p incongruent roots  x=x1,x2,,xn.  We then have the formal congruence


    Especially, we have


    because Fermat’s little theorem guarantees the roots  x=1, 2,,p-1  for the congruence  xp-1-10(modp).  If the value  x=0  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 p were not prime, then the number -1 would be divisible by any proper divisor of p.


  • 1 F. Stöwener: “Simultanbeweis des Fermatschen und Wilsonschen Satzes”.  – Elemente der Mathematik 30 (1975).
Title formal congruence
Canonical name FormalCongruence
Date of creation 2013-03-22 14:23:50
Last modified on 2013-03-22 14:23:50
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 32
Author pahio (2872)
Entry type Definition
Classification msc 11A07
Classification msc 11C08
Synonym polynomial congruence
Related topic CongruenceRelationOnAnAlgebraicSystem
Related topic APolynomialOfDegreeNOverAFieldHasAtMostNRoots
Related topic IdealDecompositionInDedekindDomain
Related topic ExampleOfGcd
Related topic FermatsLittleTheorem
Related topic WilsonsTheorem
Related topic FreshmansDream
Related topic UsingPrimitiveRootsAndIndexToSolveCongruences
Related topic SufficientConditio
Defines formally congruent
Defines identical congruence