Two-term congruence

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binomial congruence, power congruence

An algebraic congruence of the form


where are coprime integers and is a natural number. If the congruence (1) is solvable, is called an -th power residue modulo ; otherwise, is an -th power non-residue modulo .

The problem of solvability of a two-term congruence to a composite modulus can be reduced to the study of the corresponding problem to a prime modulus (cf. Congruence (in algebra)). For the prime modulus power congruence problem there is a solvability criterion, due to L. Euler: For the congruence

to be solvable, it is necessary that

where is the greatest common divisor of the numbers and ; if this condition is met, the congruence in question has exactly solutions.

It follows immediately from Euler's criterion that the numbers have exactly -th power residues and non-residues modulo .

The converse problem is much more complicated: To find all moduli to which a given number is a residue (or a non-residue) of power . It was established by Euler that the solvability or non-solvability of the congruence () depends on whether or not the prime modulus forms part of certain arithmetical progressions. A rigorous proof of this result was given first by C.F. Gauss in 1801 (see [4] and Gauss reciprocity law; Quadratic reciprocity law). Gauss further noted that a complete solution of the problem for is only possible if the ring of rational integers is extended somewhat. Thus, in establishing the reciprocity law for biquadratic residues he was forced to extend the ring of rational integers to the ring of complex integers . The solvability or non-solvability of the biquadratic congruence () in the ring for a given will depend on the value of the residue of the number to some constant modulus of the ring .

A new stage in the study of two-term congruences and their applications to other theoretical problems was initiated by I.M. Vinogradov, who showed in 1914 that the number of quadratic residues to the prime modulus among the numbers , , is given by the formula

where . A similar result was subsequently obtained by Vinogradov for a more general problem on the number of solutions of the congruence

when runs through an incomplete system of residues .


[1] B.A. Venkov, "Elementary number theory" , Wolters-Noordhoff (1970) (Translated from Russian)
[2] I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (Translated from Russian)
[3] I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian)
[4] C.F. Gauss, "Untersuchungen über höhere Arithmetik" , A. Maser (1889) (Translated from Latin)


In [a2] it is shown that the smallest non-quadratic residue modulo a prime is smaller than for any .


[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. 6
[a2] D.A. Burgess, "The distribution of quadratic residues and non-residues" Mathematica , 4 (1957) pp. 106–112
How to Cite This Entry:
Two-term congruence. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article