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Residue


The word residue is used in a number of different contexts in mathematics. Two of the most common uses are the complex residue of a pole, and the remainder of a congruence.

The number b in the congruence a=b (mod m) is called the residue of a (mod m). The residue of large numbers can be computed quickly using congruences. For example, to find 37^(13) (mod 17), note that

37=3
(1)
37^2=3^2=9=-8
(2)
37^4=81=-4
(3)
37^8=16=-1,
(4)

so

 37^(13)=37^(1+4+8)=3(-4)(-1)=12 (mod 17).
(5)

See also

Biquadratic Residue, Common Residue, Complete Residue System, Complex Residue, Congruence, Cubic Residue, Minimal Residue, Multiplicative Order, Quadratic Residue, Residue Class, Residue Theorem

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References

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 55-56, 1993.

Referenced on Wolfram|Alpha

Residue

Cite this as:

Weisstein, Eric W. "Residue." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Residue.html

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