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Advanced Algorithms and Complexity :
Lecture 22
Congruences
October 1, 2018
1 Congruences
If an integer m 6 = 0 divides the difference a − b, we say that a is congruent to b modulo m and write a ≡ b( mod m). If a − b is not divisible by m, we say that a is not congruent to b modulo m, and in this case we write a a 6 ≡ b( mod m). Let a, b, c, d denote integers. Then:
- a ≡ b( mod m), b ≡ a( mod m) and a − b ≡ 0( mod m) are equiva- lent statements.
- If a ≡ b( mod m) and c ≡ d( mod m), then a + c ≡ b + d( mod m)
- If a ≡ b( mod m) and c ≡ d( mod m), then ac ≡ bd( mod m)
- If a ≡ b( mod m) and d|m, d > 0, then a ≡ b( mod d)
- If a ≡ b( mod m) then ac ≡ bc( mod mc) for c > 0
Let f denote a polynomial with integral coefficients. If a ≡ b( mod m) then f (a) ≡ f (b)( mod m). If x ≡ y( mod m) then y is called a residue of x modulo m.
A set x 1 , x 2 , ....xm is called a complete residue system modulo m if for every integer y there is one and only one xj such that y ≡ xj ( mod m).
If b ≡ c( mod m), then (b, m) = (c, m). Let b = c + mx, then (b, m) = (c + mx, m) = (c + mx − mx, m) = (c, m).
A reduced residue system modulo m is a set of integers ri such that (ri, m) = 1, ri 6 ≡ rj ( mod m) if i 6 = j and such that every x prime to m is congruent modulo m to some member ri of the set. All reduced residue systems modulo m will contain the same number of members, a number that is denoted by φ(m). This function is called Euler’s φ-function, sometimes the totient. The number φ(m) is the number of positive integers less than or equal to m that are relatively prime to m.
Let (a, m) = 1. Let r 1 , r 2 , r 3 , ...., rn be a complete, or a reduced residue system modulo m. The ar 1 , ar 2 , ...., arn is a complete, or a reduced, residue system, respectively, modulo m. (ri, m) = 1 =⇒ (ari, m) = 1 ri ≡ rj ( mod m) =⇒ ari ≡ arj ( mod m) ari ≡ arj ( mod m) =⇒ ri ≡ rj ( mod m) since (a, m) = 1
2 Fermat’s Little Theorem
Let p denote a prime. If a is not divisible by p then ap−^1 ≡ 1( mod p). For every integer a, ap^ ≡ a( mod p).
3 Euler’s Theorem
If (a, m) = 1, then aφ(m)^ ≡ 1( mod m)
Let r 1 , r 2 , ...., rφ(m) be a reduced residue system modulo m. Then ar 1 , ar 2 , ...., arφ(m), is also a reduced residue system modulo m. =⇒ ari ≡ rj ( mod m) for unique j. =⇒ aφ(m)Πri ≡ Πrj ( mod m) aφ(m)^ ≡ 1( mod m)
If (a, m) = 1 then there is an x such that ax ≡ 1( mod m). Any two
