Write your own answers 10. Let p be a prime number, and let a be an...
please prove proofs and do 7.4 7.2 Theorem. Let p be a prime, and let b and e be integers. Then there exists a linear change of variahle, yx+ with a an integer truns- farming the congruence xbx e0 (mod p) into a congruence of the farm y (mod p) for some integer 8 Our goal is to understand which integers are perfect squares of other inte- gers modulo a prime p. The first theorem below tells us that half...
Let p be an odd prime. Write p in the form p = 2k + 1 for some k E N. Prove that kl-(-1)* mod p. Hint: Each j e Z satisfies j (p-od p.
Let p be a prime number. Prove that if there exists a solution to the congruence c(mod p), then there exist integers m, n such that p = m2 + 2n2. Hint. Make a careful study of the proof of Fermat's Two Square Theorem, and then try to modify that proof (or at least some portion of it) to come up with a proof of this statement. Toward the end of the proof, the following observation can be helpful: If...
ame: . (10 points) Let p > 3 be any prime number. (a) Show that p mod 6 is equal to 1 or 5 (b) Use part (a) to prove that pe - 1 is always a multiple of 24.
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...
8.15 Lemma. Let p be a prime and let a be a natural mumber not divisible by p. Then there exist integers x and y such that ax y (mod p) with 0xl.lylP 8.16 Theorem. Ler p be a prime such that p 1 (mod 4). Then p is equal to the sum of two squares of natural numbers. (int: Iry applying the previous lemma to a square root of- mochdo p.) Knowing which primes can be written as the...
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 +...
Let p be a prime number. Prove that 19–1 + 2P-1 + ....(p – 1)p-1 = -1 mod p
4.3. Let p 2 3 be a prime, and let m 2 1 be an integer that is relatively prime to p 1. (a) Prove that the map to itself. (b) Prove that the equation is an isomorphism of F has exactly p 1 projective solutions with x, y,zEF
Problem 6: Let p be an odd prime number, so that p= 2k +1 for some positive integer k. Prove that (k!)2 = (-1)k+1 mod p. Hint: Try to see how to group the terms in the product (p − 1)! = (2k)! = 1 * 2 * 3... (2k – 2) * (2k – 1) * (2k) to get two products, each equal to k! modulo p.