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22. Suppose that (ab, p)- 1 and that p> 2. Show that the number of solutions (x, y) of the congru...
1. Show that the number of solutions (x mod p, y mod p) to the equation...
1. Show that the number of solutions (x mod p, y mod p) to the equation x² + 1 = y2 mod p is p- p (6+1) k=0
4. Suppose that p is a prime of the form 8k + 1 . Show that the congruence x4 has ether 0 solutio...
4. Suppose that p is a prime of the form 8k + 1 . Show that the congruence x4 has ether 0 solutions or 4 solutions. 2 (mod P)
4. Suppose that p is a prime of the form 8k + 1 . Show that the congruence x4 has ether 0 solutions or 4 solutions. 2 (mod P)
39. Suppose that the polynomial congruence f(x)0 (mod 7) has two distinct so- 0 lutions, what are the possible number of solutions of the congruence f(x) (mod 49)? 39. Suppose that the polyn...
39. Suppose that the polynomial congruence f(x)0 (mod 7) has two distinct so- 0 lutions, what are the possible number of solutions of the congruence f(x) (mod 49)?
39. Suppose that the polynomial congruence f(x)0 (mod 7) has two distinct so- 0 lutions, what are the possible number of solutions of the congruence f(x) (mod 49)?
2. Let p be an odd prime. We saw last week that the problem of counting solutions to the congruen...
2. Let p be an odd prime. We saw last week that the problem of counting solutions to the congruence (mod p) is only interesting when p has the form 4k1. For the rest of this problem let p 4k+1. (a) Show that (mod -1 5 (mod 8) (b) Show that pEl (mod 8 -1 p5 (mod 8) (c) Draw condlusions about the number of solutions to these congruences 14 (mod p) -1 (mod p) (mod p)
2. Let p...
I need help with number 3 on my number theory hw. Exercise 1. Figure out how...
I need help with number 3 on my number theory
hw.
Exercise 1. Figure out how many solutions x2 = x (mod n) has for n = 5,6,7, and then compute how many solutions there are modulo 210. Exercise 2. (a) Find all solutions to x2 +8 = 0 (mod 11). (b) Using your answer to part (a) and Hensel's Lemma, find all solutions to x2 +8 = 0 (mod 121). Exercise 3. Solve f(x) = x3 – x2 +...
1. (Complex Multiplication) Let E : y x3 y23 to this congruence mod p. So for...
1. (Complex Multiplication) Let E : y x3 y23 to this congruence mod p. So for example, #E(Z3) = 3 because we have the solutions (0, 0), (1,0) and (2,0) and no more. - x. Then we can reduce E mod p to get mod p for various primes p. We write #E(Z») for the number of solutions This particular equation has some miraculous explore here patterns we (a) Make a chart that lists p, #E(Zp), and #E(Z) - p...
Please solve the above 4 questions. 1. Using the extended Euclidean Algorithm, find all solutions of the linear congrue...
Please solve the above 4 questions.
1. Using the extended Euclidean Algorithm, find all solutions of the linear congruence 217x 133 (mod 329), where 0 x < 329 (Eg. if 5n, n 0,. ,6) 24 + 5n, п %3D 0, 1, . .., 6, type 24 + x< 11 2. Find all solutions of the congruence 7x = 5 (mod 11) where 0 (Eg. if 4,7 10, 13, type 4,7,10,13, none. or if there are no solutions, type I 3....
You may use the following facts to answer the questions below Fact 1: Suppose that Xi. . . . , X, are independent and X.* GAM (θ.k.) for -1 -1 Fact 2: If Y GAM(0,n aYGAM(ab,n) for any number a &g...
You may use the following facts to answer the questions below Fact 1: Suppose that Xi. . . . , X, are independent and X.* GAM (θ.k.) for -1 -1 Fact 2: If Y GAM(0,n aYGAM(ab,n) for any number a >0 1. Suppose that V-GAM(1m) and let lPa θν, where θ > 0. (a) Show that, for any given positive number a, P> a) is an increasing function of (b) What is the probability distribution of W? (c) Would you...
2. Suppose X and Y are independent continuous random variables. Show that P(Y < X) =...
2. Suppose X and Y are independent continuous random variables. Show that P(Y < X) = | Fy(x) · fx (x) dx -oo where Fy is the CDF of Y and fx is the PDF of X [hint: P[Y E A] = S.P(Y E A|X = x) · fx(x) dx]. Rewrite the above equation as an expectation of a function of X, i.e. P(Y < X) = Ex[•]. Use the above relation to compute P[Y < X] if X~Exp (2)...
7. Suppose X ~N(3, 22) (1) Evaluate P (2 Xs5), P-4<Xs10), P>2) (2) Decide C so...
7. Suppose X ~N(3, 22) (1) Evaluate P (2 Xs5), P-4<Xs10), P>2) (2) Decide C so that P (X> C) P (sc) Suppose the density function of X is 04 x)8 0, else Find the density function of Y-2X+8.
1. Show that the number of solutions (x mod p, y mod p) to the equation x² + 1 = y2 mod p is p- p (6+1) k=0
4. Suppose that p is a prime of the form 8k + 1 . Show that the congruence x4 has ether 0 solutions or 4 solutions. 2 (mod P)
4. Suppose that p is a prime of the form 8k + 1 . Show that the congruence x4 has ether 0 solutions or 4 solutions. 2 (mod P)
39. Suppose that the polynomial congruence f(x)0 (mod 7) has two distinct so- 0 lutions, what are the possible number of solutions of the congruence f(x) (mod 49)?
39. Suppose that the polynomial congruence f(x)0 (mod 7) has two distinct so- 0 lutions, what are the possible number of solutions of the congruence f(x) (mod 49)?
2. Let p be an odd prime. We saw last week that the problem of counting solutions to the congruence (mod p) is only interesting when p has the form 4k1. For the rest of this problem let p 4k+1. (a) Show that (mod -1 5 (mod 8) (b) Show that pEl (mod 8 -1 p5 (mod 8) (c) Draw condlusions about the number of solutions to these congruences 14 (mod p) -1 (mod p) (mod p)
2. Let p...
I need help with number 3 on my number theory
hw.
Exercise 1. Figure out how many solutions x2 = x (mod n) has for n = 5,6,7, and then compute how many solutions there are modulo 210. Exercise 2. (a) Find all solutions to x2 +8 = 0 (mod 11). (b) Using your answer to part (a) and Hensel's Lemma, find all solutions to x2 +8 = 0 (mod 121). Exercise 3. Solve f(x) = x3 – x2 +...
1. (Complex Multiplication) Let E : y x3 y23 to this congruence mod p. So for example, #E(Z3) = 3 because we have the solutions (0, 0), (1,0) and (2,0) and no more. - x. Then we can reduce E mod p to get mod p for various primes p. We write #E(Z») for the number of solutions This particular equation has some miraculous explore here patterns we (a) Make a chart that lists p, #E(Zp), and #E(Z) - p...
Please solve the above 4 questions.
1. Using the extended Euclidean Algorithm, find all solutions of the linear congruence 217x 133 (mod 329), where 0 x < 329 (Eg. if 5n, n 0,. ,6) 24 + 5n, п %3D 0, 1, . .., 6, type 24 + x< 11 2. Find all solutions of the congruence 7x = 5 (mod 11) where 0 (Eg. if 4,7 10, 13, type 4,7,10,13, none. or if there are no solutions, type I 3....
You may use the following facts to answer the questions below Fact 1: Suppose that Xi. . . . , X, are independent and X.* GAM (θ.k.) for -1 -1 Fact 2: If Y GAM(0,n aYGAM(ab,n) for any number a >0 1. Suppose that V-GAM(1m) and let lPa θν, where θ > 0. (a) Show that, for any given positive number a, P> a) is an increasing function of (b) What is the probability distribution of W? (c) Would you...
2. Suppose X and Y are independent continuous random variables. Show that P(Y < X) = | Fy(x) · fx (x) dx -oo where Fy is the CDF of Y and fx is the PDF of X [hint: P[Y E A] = S.P(Y E A|X = x) · fx(x) dx]. Rewrite the above equation as an expectation of a function of X, i.e. P(Y < X) = Ex[•]. Use the above relation to compute P[Y < X] if X~Exp (2)...
7. Suppose X ~N(3, 22) (1) Evaluate P (2 Xs5), P-4<Xs10), P>2) (2) Decide C so that P (X> C) P (sc) Suppose the density function of X is 04 x)8 0, else Find the density function of Y-2X+8.
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