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c - CONGRUENCE The existence of solutions to binary...

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CONGRUENCE The existence of solutions to binary quadratic equations depends in general on the existence of an integer u such that a quadratic expression a 2 u 2 + a 1 u + a 0 (with integer coefficients) is divisible by a fixed integer n. We now examine the problem of finding values of u (if any) such that a general polynomial f(u) is divisible by an integer n. To illustrate the application of the various theorems we use them to examine the representation of integers by sums of two squares; that is, solutions to the binary quadratic equation x 2 + y 2 = n. If two integers a and b have the same principal remainder on division by n, or equivalently a - b is divisible by n, we say that a is congruent with b, modulo n and write a ≡ b (mod n). In this notation, which was introduced by Gauss in Disquisitiones Arithmeticae the problem we are addressing is the solution of the congruence f(u) ≡ 0 (mod n). Theorem 1 : If a ≡ b (mod n) and c ≡ d (mod n) then a + c ≡ b + d (mod n) and ac bd (mod n) Proof: If a ≡ b (mod n) a nd c ≡ d (mod n) then a =b + k 1 n and c = d +k 2 n so a + c = b + d + (k 1 + k 2 )n and ac = bd + (k 1 d + k 2 b + k 1 k 2 n)n ¤ The relationship of congruence (mod n) partitions the integers into n mutually exclusive congruence classes , with each class consisting of those integers having the same principal remainder on division by n. If we denote as n Z the set of all integral multiples of n, then the set n Z + k is the congruence class containing k. We can treat each congruence class as an algebraic entity, with addition and multiplication operations defined by: (n Z + a i ) + (n Z + a j ) = n Z + (a i + a j ) and (n Z + a i )·(n Z + a j ) = n Z + a i a j respectively. We refer to this entity as Z /n Z . It is clear that, by definition, a i ≡ a j (mod n) iff n Z + a i = n Z + a j . In this notation, our problem is one of solving f(n Z + u) = n Z + 0. We shall see later (Theorem 3) that the coefficients of the polynomial may be expressed either as integers or as elements of Z /n Z . We now examine the properties of addition in Z /n Z , using Z /5 Z and Z /6 Z as examples. In the tables, we abbreviate n Z + a as a where a lies in the interval [0, n-1]. Z /5 Z ,+ 0 1 2 3 4 Z/ 6 Z ,+ 0 1 2 3 4 5 0 0 1 2 3 4 0 0 1 2 3 4 5 1 1 2 3 4 0 1 1 2 3 4 5 0 2 2 3 4 0 1 2 2 3 4 5 0 1 3 3 4 0 1 2 3 3 4 5 0 1 2 4 4 0 1 2 3 4 4 5 0 1 2 3 5 5 0 1 2 3 4 Z /n Z , + is an example of an abstract algebraic entity called a commutative group. A commutative group consists of a (finite or infinite) set of elements and a binary operation (which we can denote as ◦) that combines two elements to form another element and has the following properties: closure : if a,b are in a group G then a ◦b is in G; identity : there is an element i in G such that i ◦a = a for every a in G; associative : (a ◦b)◦ c = a ( b ◦c); inverse : for each element a there is an inverse element a' such that a ◦a' =i; and commutative : a ◦b = b◦a. It can be shown (Supplement, 1) that the identity and inverse are necessarily unique.
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CONGRUENCE 2 Z , Q and R are groups with respect to the operation of addition, with the identity element being zero and the inverse of a being -a. Z /n Z
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