Reduced residue system

In mathematics, a subset R of the integers is called a reduced residue system modulo n if:

  1. gcd(r, n) = 1 for each r in R,
  2. R contains φ(n) elements,
  3. no two elements of R are congruent modulo n.[1][2]

Here φ denotes Euler's totient function.

A reduced residue system modulo n can be formed from a complete residue system modulo n by removing all integers not relatively prime to n. For example, a complete residue system modulo 12 is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. The so-called totatives 1, 5, 7 and 11 are the only integers in this set which are relatively prime to 12, and so the corresponding reduced residue system modulo 12 is {1, 5, 7, 11}. The cardinality of this set can be calculated with the totient function: φ(12) = 4. Some other reduced residue systems modulo 12 are:

  • {13,17,19,23}
  • {−11,−7,−5,−1}
  • {−7,−13,13,31}
  • {35,43,53,61}

Facts
  • If {r1, r2, ... , rφ(n)} is a reduced residue system modulo n with n > 2, then .
  • Every number in a reduced residue system modulo n is a generator for the additive group of integers modulo n.
  • If {r1, r2, ... , rφ(n)} is a reduced residue system modulo n, and a is an integer such that gcd(a, n) = 1, then {ar1, ar2, ... , arφ(n)} is also a reduced residue system modulo n.[3][4]

See also

Notes
  1. Long (1972, p. 85)
  2. Pettofrezzo & Byrkit (1970, p. 104)
  3. Long (1972, p. 86)
  4. Pettofrezzo & Byrkit (1970, p. 108)

References
  • Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77171950
  • Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 71081766
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