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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) Suppose a and m are any two integers with m not zero. We say r is a residue of a modulo m if a = r (mod m). This is the same as m divides a  r (see congruence), or a = r + qm for some integer q. The division algorithm tells us that there is a unique residue r satisfying 0 < r < m, and this remainder r is called the least nonnegative residue of a modulo m. A set of integers form a complete system of residues modulo m if every integer is congruent modulo m to exactly one integer in the set. So a complete system of residues includes exactly one element from each congruence class modulo m. For example, if m is positive, then {0, 1, 2, 3,..., m1}is a complete system of residues (called the least nonnegative residues modulo m). If m is positive and odd, then we sometimes use the system {  (m1)/2,  (m3)/2, ..., 1, 0, 1, ..., (m3)/2, (m1)/2}There are infinitely many complete residue systems for each modulus m.
Chris K. Caldwell © 19992019 (all rights reserved)
