If there is an Integer such that

 (1)

then is said to be a quadratic residue (mod ). If not, is said to be a quadratic nonresidue (mod ). For example, , so 6 is a quadratic residue (mod 10). The entire set of quadratic residues (mod 10) are given by 1, 4, 5, 6, and 9, since

making the numbers 2, 3, 7, and 8 the quadratic nonresidues (mod 10).

A list of quadratic residues for is given below (Sloane's A046071), with those numbers not in the list being quadratic nonresidues of .

 Quadratic Residues 1 (none) 2 1 3 1 4 1 5 1, 4 6 1, 3, 4 7 1, 2, 4 8 1, 4 9 1, 4, 7 10 1, 4, 5, 6, 9 11 1, 3, 4, 5, 9 12 1, 4, 9 13 1, 3, 4, 9, 10, 12 14 1, 2, 4, 7, 8, 9, 11 15 1, 4, 6, 9, 10 16 1, 4, 9 17 1, 2, 4, 8, 9, 13, 15, 16 18 1, 4, 7, 9, 10, 13, 16 19 1, 4, 5, 6, 7, 9, 11, 16, 17 20 1, 4, 5, 9, 16

Given an Odd Prime and an Integer , then the Legendre Symbol is given by

 (2)

If
 (3)

then is a quadratic residue (+) or nonresidue (). This can be seen since if is a quadratic residue of , then there exists a square such that , so
 (4)

and is congruent to 1 (mod ) by Fermat's Little Theorem. is given by
 (5)

More generally, let be a quadratic residue modulo an Odd Prime . Choose such that the Legendre Symbol . Then defining

 (6) (7) (8)

gives
 (9) (10)

and a solution to the quadratic Congruence is
 (11)

The following table gives the Primes which have a given number as a quadratic residue.

 Primes 2 3 5 6

Finding the Continued Fraction of a Square Root and using the relationship

 (12)

for the th Convergent gives
 (13)

Therefore, is a quadratic residue of . But since , is a quadratic residue, as must be . But since is a quadratic residue, so is , and we see that are all quadratic residues of . This method is not guaranteed to produce all quadratic residues, but can often produce several small ones in the case of large , enabling to be factored.

The number of Squares in is related to the number of quadratic residues in by

 (14)

for (Stangl 1996). Both and are Multiplicative Functions.

References

Burton, D. M. Elementary Number Theory, 4th ed. New York: McGraw-Hill, p. 201, 1997.

Courant, R. and Robbins, H. Quadratic Residues.'' §2.3 in Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 38-40, 1996.

Guy, R. K. Quadratic Residues. Schur's Conjecture'' and Patterns of Quadratic Residues.'' §F5 and F6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 244-248, 1994.

Niven, I. and Zuckerman, H. An Introduction to the Theory of Numbers, 4th ed. New York: Wiley, p. 84, 1980.

Rosen, K. H. Ch. 9 in Elementary Number Theory and Its Applications, 3rd ed. Reading, MA: Addison-Wesley, 1993.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 63-66, 1993.

Sloane, N. J. A. Sequence A046071 in The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Stangl, W. D. Counting Squares in .'' Math. Mag. 69, 285-289, 1996.

Wagon, S. Quadratic Residues.'' §9.2 in Mathematica in Action. New York: W. H. Freeman, pp. 292-296, 1991.