3 Solutions of congruences

Transcription

1 3 Solutions of congruences 3.1 The Chinese Remainder Theorem revisited Let f(x) be a polynomial with coefficients in Z. A congruence equation is of the form f(x) 0 (mod m), where m is a positive integer. In this Chapter, we will study such equations. theorem 3.1 Let f(x) be a fixed polynomial with integral coefficients, and for anypositiveintegermletn(m)denotethenumberofsolutionsofthecongruence f(x) 0 (mod m). If m = m 1 m 2 where (m 1,m 2 ) = 1, then N(m) = N(m 1 )N(m 2 ). If m = p pαp is the canonical factorization of m, then N(m) = p N(p αp ). Let m be a positive integer. Suppose (m 1,m 2 ) = 1 and that there are N(m 1 ) solutions, say {a 1,,a N(m1)}, to f(x) 0 (mod m 1 ) and N(m 2 ) solutions, say {b 1,,b N(m2)}, to the equation f(x) 0 (mod m 2 ). To each pair of solutions (a i,b j ) there exists an integer c ij modulo m such that f(c ij ) 0 (mod m 1 ) and f(c ij ) 0 (mod m 2 ). The existence of c ij is guaranteed by the Chinese Remainder Theorem. This implies that f(c ij ) 0 (mod m)

2 36 Solutions of congruences and therefore Next, if N(m 1 )N(m 2 ) N(m). f(x) 0 (mod m) and m = m 1 m 2,(m 1,m 2 ) = 1, then by Chinese Remainder Theorem, there are N(m) pairs (a mod m 1,b (mod m 2 )) such that f(a) 0 (mod m 1 ) and This implies that Hence, f(b) 0 (mod m 2 ). N(m) N(m 1 )N(m 2 ). N(m 1 m 2 ) = N(m 1 )N(m 2 ). example 3.1 Let f(x) = x 2 +x+3. Find all roots of the congruence f(x) 0 (mod 15). Solutions. The solutions for f(x) 0 (mod 15) are 3,6,8,11. The solutions for f(x) 0 (mod 3) are 0 and 2 and for f(x) 0 (mod 5) are 1 and 3. Note that N(15) = N(3)N(5). 3.2 Prime power moduli Theorem 3.1 shows that in order to solve f(x) 0 (mod m), it suffices to solve f(x) 0 (mod p α ) (3.1) for p α m where p is a prime. We now show that in order to solve (3.1), if suffices to find the solutions of By Taylor s series expansion, f(x) 0 (mod p). f(a+tp j ) = f(a)+tp j f (a)+t 2 p 2j f (a)/2+ +t n p nj f (n) (a)/n!, (3.2)

3 3.2 Prime power moduli 37 where n is the degree of f(x). Observe that if then Therefore, f (k) (x) k! f(x) = = n c r x r, r=0 n ( ) r c r x r k. k r=k f (k) (a) k! is an integer for 0 k n. We now suppose that Since f(a) 0 (mod p j ). t k p kjf(k) (a) k! 0 (mod p j+1 ) for k > 1 (2j > j for j 1), we conclude from (3.2) that f(a+tp j ) f(a)+tp j f (a) (mod p j+1 ). (3.3) We now return to (3.3). We see that we need to choose We now split our investigation into cases: Case 1. p f (a). tf (a) f(a) p j (mod p). (3.4) In this case, there is a unique solution t for (3.4) and hence we obtain theorem 3.2 (Hensel s Lemma) Suppose that f(x) is a polynomial with integral coefficient. If f(a) 0 (mod p j ) and f (a) 0 (mod p), then there is a unique t (mod p) such that f(a+tp j ) 0 (mod p j+1 ). Case 2. If p f (a) but p f(a) p j then (3.4) is not solvable. Case 3. If p f (a) and p f(a) p j then there are p solutions for t in (3.4).

4 38 Solutions of congruences Remark 3.3 Note that if u is solution of f(x) 0 (mod p j+1 ), (3.5) then by the fact that we can always write u = a+tp j for some a and t where 0 a < p j and 0 t p, we conclude that all solutions of (3.5) are of the form a+tp j, with f(a) 0 (mod p j ), (3.6) and so, we have exhausted all possible solutions to (3.5) by considering solutions of (3.6). example 3.2 Solve x 3 2x 2 +3x+9 0 (mod 3 3 ). Solutions. The solutions to the congruence f(x) 0 (mod 3) are x = 0 and 2. Now, f (x) = 3x 2 4x+3. Since f (2) 0 (mod 3), we seethat there is aunique solutionarisingfor x = 2. We need to solve 7t 15/3 (mod 3) and it turns out that t = 1. The solution for the congruence f(x) 0 (mod p) arising from x = 2 is therefore 2+3 = 5. Now Hence there are three solutions for f (0) = 3 and f(0) = 9. f(x) 0 (mod 9) corresponding to the solution x = 0 from These are x = 0,3 and 6. We now have four solutions for f(x) 0 (mod 3). f(x) 0 (mod 3 2 ). We check that f(0) = 9 and f (0) = 3 and so f(x) 0 (mod 27) has no solution arising from x = 0. Next, f(3) = 27 and f (3) = 18 and so, there are three solutions arising from x = 3. These are x = 3,12 and 21.

5 3.3 Prime moduli 39 For x = 6 we have f(6) = 171 and f (6) = 81. But 3 (171/9) and hence, there are no solutions arising from x = 6. For x = 5 we find that f(5) = 99 and f (5) = 58 and we need to solve the congruence 58t 11 (mod 3). The solution is t = 1 and so 14 is the solution for the congruence. In conclusion the solutions to the congruence are 3,12,14 and 21. f(x) 0 (mod 27) 3.3 Prime moduli In the previous section, we have seen that we can reduce the problem of finding solutions for f(x) 0 (mod p α ) to finding solutions for f(x) 0 (mod p). We will first prove a result for polynomials over a field F, that is an analogue for the Division Algorithm for integers. theorem 3.4 Given the polynomials f(x),g(x) F[x], where deg g(x) > 0, there exist polynomials q(x), r(x) F[x] such that f(x) = g(x)q(x)+r(x), with either r(x) = 0 or 0 degr(x) < degg(x). We proceed by induction on deg f(x). First fix the polynomial g(x). Suppose f(x) = 0 or deg f(x) < deg g(x), then f(x) = 0 g(x)+f(x). So we may assume that deg f(x) deg g(x). Suppose the statement is true for any polynomials with degrees less than or equal to n 1. Let f(x) be a polynomial of degree n. Write f(x) = a 0 +a 1 x+ +a n x n, and g(x) = b 0 +b 1 x+ +b m x m, with a n 0 and b m 0 and n m. Since F is a field, b 1 m polynomial P(x) := f(x) b 1 m a nx n m g(x) has degree n 1. Hence, there exist q(x) and r(x) such that ( ) exists and the P(x) = q(x)g(x)+r(x),

6 40 Solutions of congruences with deg r(x) = 0 or deg r(x) < deg g(x). But by (*), hence the result. f(x) = (q(x)+b 1 m a nx n m )g(x)+r(x), We now let F be the finite field Z/pZ. Let g(x) be the polynomial x p x and suppose the degree of f(x) is n p. By Theorem 3.4, we conclude that f(x) = (x p x)q(x)+r(x), with q(x),r(x) (Z/pZ)[x] and 0 degr(x) < p. Suppose u is such that f(u) 0 (mod p). By Fermat s Little Theorem, we find that u p u 0 (mod p). Therefore, r(u) 0 (mod p).conversely,ifr(u) 0 (mod p),thenf(u) 0 (mod p).thisshows that to study f(x) 0 (mod p), it suffices to consider those polynomial f(x) with degree of f(x) is less than p. theorem 3.5 (Lagrange) Let f(x) be a polynomial of degree n < p and a n 0 (mod p). Then the congruence has at most n solutions. f(x) 0 (mod p) (3.7) We prove this by induction on the degree of f(x). If n = 0, a 0 0 (mod p) implies that there are no solution to (3.7). If the degree of f(x) is 1, then we see that we are solving a 1 x+a 2 0 (mod p). This is solvable since a 1 0 (mod p). Therefore the number of solution is 1. Suppose the result holds for all polynomials of degree less than n. Let f(x) be a polynomial of degree n. If f(x) 0 (mod p) has no solution, then we are done. Next, suppose u be a root of the nth degree polynomial f. Then by the division algorithm for Q[x], f(x) = (x u)g(x)+r 0 for some polynomial g(x) of degree n 1. Hence, r 0 0 (mod p) since Therefore, f(u) 0 (mod p) and (u u)g(u) 0 (mod p). f(x) (x u)g(x) (mod p).

7 3.3 Prime moduli 41 If b is a root of f(x) 0 (mod p), then either p (b u) or p g(b), by Euclid s lemma. If p g(b), then g(b) 0 (mod p) and by induction there are at most n 1 possible values for b. Therefore, we conclude that f(x) can have at most n roots. The above result is false if p is not a prime. For example the congruence equation has four solutions x = 1,3,5,7. x 2 1 (mod 8) corollary 3.6 If d (p 1) then the congruence x d 1 (mod p) has exactly d solutions. By previous theorem, the congruence cannot have more than d solutions. For the converse, let p 1 = de. Note that x p 1 1 = (x d 1)(x d(e 1) +x d(e 2) + +x d +1). By the previous theorem, the polynomial x d(e 1) +x d(e 2) + +x d +1 cannot have more than de d or p 1 d solutions. Hence, if x d 1 has less than d solutions then x p 1 1 must have less than p 1 d + d = p 1 solutions. But there are exactly p 1 solutions to the equation x p 1 1 by Fermat s little theorem. This contradiction shows that x d 1 must have exactly d solutions. example 3.3 The congruence x 10 1 (mod 811) has ten solutions and these are 1,212,241,311,339,472,500,570,599 and 810. Remark 3.7 In general,if d (p 1), the number of solutions for the congruence x d 1 (mod p) is (d,p 1). To see this, we need the fact that the multiplicative group of Z/pZ is cyclic, say, generated by g. This fact will be proved in the next Chapter. Now, if u is a solution to u d 1 (mod p), then u = g y for some integer y since g generates (Z/pZ). Therefore, g yd 1 (mod p). But since the order of g is p 1, we conclude that p 1 divides yd. This implies that yd 0 (mod p 1) and the number of solutions to this congruence is (d,p 1) by Theorem 1.40.

8 42 Solutions of congruences theorem 3.8 (Wilson) If p is prime then (p 1)! 1 (mod p). If p = 2, then 1 1 (mod 2). Let p be an odd prime. By Fermat s little theorem, we know that the polynomial x p 1 1 (x 1)(x 2) (x (p 1)) has at least p 1 roots. Since the degree of the above polynomial is p 2 and such polynomial can have at most p 2 roots, we conclude that the polynomial must be identically 0 modulo p. Setting x = 0 we conclude that since p is odd. (p 1)! 1 (mod p) Remark 3.9 Wilson s Theorem is false when the prime p is replaced by a composite number m. Suppose m is composite. Let m = k p α k k. If m is not a prime power, then and so m 1 = p α1 1 m 1 > p α1 1 (m 1)! 0 (mod p α1 1 ). If m is a prime power, then let m = p α. If α > 2, then m 1 > p α 1 and p and p α 1 are distinct divisors of (m 1)! and hence (m 1)! 0 (mod p α ). It remains to show the result for α = 2, or when m = p 2. In this case m 1 = p 2 1 > p. The only point we have to worry is that m 1 < 2p. For if m 1 2p then both p and 2p are distinct divisors of (m 1)! and hence, (m 1)! 0 (mod p 2 ). But if p 2 1 < 2p we have p(p 2) < 1 and this is only possible if p = 2. We have thus prove that (m 1)! 0 (mod m) if m > 4 and composite. For p = 2 and α = 2 we have 3! 2 1 (mod 4). Hence, we find that if and only if m is a prime. (m 1)! 1 (mod m)

9 3.4 Wolstenholme s congruence Wolstenholme s congruence Let F(x) = (x 1)(x 2) (x (p 1)). Write F(x) = x p 1 σ 1 x p 2 +σ 2 x p 3 +σ p 3 x 2 σ p 2 x+σ p 1, where σ j denotes the sum of all products of j distinct roots of F(x). In the proof of Theorem 3.8, we have seen that the polynomial f(x) = x p 1 1 (x 1)(x 2) (x (p 1)) = x p 1 1 F(x) is the zero polynomial in (Z/pZ)[x]. This means that in (Z/pZ)[x], x p 1 1 = F(x) = (x 1)(x 2) (x (p 1)) = x p 1 σ 1 x p 2 +σ 2 x p 3 +σ p 3 x 2 σ p 2 x+σ p 1. Comparing the coefficients of x j on both sides of the above polynomials in (Z/pZ)[x], we conclude that σ j 0 (mod p),1 j p 2 (3.8) We now prove a result stronger that (3.8) when j = p 2. theorem 3.10 (Wolstenholme s congruence) For prime p 5, Note that σ p 2 0 (mod p 2 ). F(p) = (p 1)! = p p 1 σ 1 p p 2 + σ p 2 p+(p 1)!. Now, By (3.8), p p 2 σ 1 p p 3 + +σ p 3 p σ p 2 = 0. σ p 3 0 (mod p) and since p 5, we find that σ p 2 0 (mod p 2 ).