Andrew Granville's An Introduction to Gauss's Number Theory PDF

By Andrew Granville

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5, after we have developed a little more theory. 3. Congruences for polynomials. Let Z[x] denote the set of polynomials with integer coefficients. 3. If f (x) ∈ Z[x] and a ≡ b (mod m) then f (a) ≡ f (b) (mod m). Proof. 2, and then d ak ≡ bk (mod m) for all integers k, by induction. 6. This result can be extended to polynomials in many variables. 1. Prove that if f (t) ∈ Z[t] and r, s ∈ Z then r − s divides f (r) − f (s). Therefore, for any given polynomial f (x) ∈ Z[x], the sequence f (0), f (1), f (2), f (3), .

Mk are a set of pairwise coprime positive integers. For any set of residue classes a1 (mod m1 ), a2 (mod m2 ), . . , ak (mod mk ), there exists a unique residue class x (mod m), where m = m1 m2 . . mk , such that x ≡ aj (mod mj ) for each j. Proof. We can map x (mod m) to the vector (x (mod m1 ), x (mod m2 ), . . , x (mod mk )). There are m1 m2 . . 17(a). Hence there is a suitable 1-to-1 correspondence between residue classes mod m and vectors, which implies the result. This is known as the Chinese Remainder Theorem because of the ancient Chinese practice (as discussed in Sun Tzu’s 4th century Classic Calculations) of counting the number of soldiers in a platoon by having them line up in three columns and seeing how many are left over, then in five columns and seeing how many are left over, and finally in seven columns and seeing how many are left over, etc.

1(c). Hence there are no solutions when gcd(a, b) does not divide c. On the other hand, GAUSS’S NUMBER THEORY 33 we have seen that there exists integers u, v such that au + bv = (a, b) and so if c = k(a, b) then a(ku) + b(kv) = c. Given one solution r, s to ar + bs = c we can find all other solutions by noting that if am + bn = c = ar + bs then a(m − r) = b(s − n). 2, so we can write m = r + b/(a, b) for some integer , and then n = s − a/(a, b). Note that the real solutions to ax + by = c are given by x = r + kb, y = s − ka, k ∈ R.

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An Introduction to Gauss's Number Theory by Andrew Granville


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