Let k > 3. Show that
(1) 3 has order 2^(k-2) modulo 2k .
(2) {3, -1} is a generating set for 2k .
Let k > 3. Show that (1) 3 has order 2^(k-2) modulo 2k . (2) {3, -1} is a generating set for 2k . Let k 3. Show that...
Let k > 3. Show that (1) 3 has order 2^(k-2) modulo 2^k . (2)
{3, -1} is a generating set for 2^k
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
Prove: Let k be a positive integer, and set n :=2k-1(2k – 1). Then (2k+1 – 1)2 = 8n +1 Prove: Let n be a positive integer, and let s and t be integers. Show that Hire (st) = n(s) in (t) mod n.
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.
Exercise 1: Let k 21 be a positive integer. Consider the set of ordered 2k-tuples Π = {(zi, r2k) : ri+r2k, as (ri, -.. .r2k) vary over T? , 21 < ri s 61. Which number appears most often as the sum In other words, suppose we roll 2k six-sided standard dice, and add up the numbers that appear. Which number (as a function of k) appears most often as the sum of the values on top?
3. Let p>3 be an odd prime and let {ri,r2, .r} be the set of incongruent primitive roots modulo p. Compute the product rir .r modulo p. Recall the proof of Wil- son's Theorem for inspiration
3. Let p>3 be an odd prime and let {ri,r2, .r} be the set of incongruent primitive roots modulo p. Compute the product rir .r modulo p. Recall the proof of Wil- son's Theorem for inspiration
(b) Let p be a prime that is congruent to 3 modulo 4. Let b ∈ Z. Let a = b (p+1)/4 . Show that a 2 ≡ ±b (mod p). (c) Give an algorithm to compute square roots of something modulo p, when p ≡ 3 (mod 4). Note: Not all things are square modulo p, so the algorithm should return the square root or inform you there is no square.
1 10 onvelge a636lutely, converges conditionally, or diverges. Justify your answer, including naming the convergence test you use. (1n(b) n7/3-4 (2k)! n-2 k-0 (-1)k 2k 4. (a) (10) Let* Find a power series for h(), and find the radius of convergence Ri for h'(x). Find the smallest reasonable positive integer n so that - (b) (10) Let A- differs from A by less than 0.01. Give reasons. 5. (a) (10) Let g(x) sin z. Write down the Taylor series for...
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrutic residue modulo pif and only ifp-a is a quadratic non-residue modulo p. 7.24 Theorem. Let p be a prime of the form p odd prime. Then p 3 (mod 4). 241 where q is an The next theorem describes the symmetry between primitive roots and quadratic residues for primes arising from odd Sophie...
2. Using a z-transform table, show that a) 2k+14[k – 1] +ek-[k] 2+ ane) b) kyku[k – 1] Hint: Express 1 k-1) in terms of u[k]. c) (2-cos(k)]u[k –1] 2-0.52+0.25 d) k(k-1) (k - 2)24-34[k - m e for m=0,1,2 or 3
Let →a=2→i−5→j−2→ka→=2i→-5j→-2k→ and →b=5→i−→kb→=5i→-k→. Find
−→a+→b-a→+b→.
Let ā = 27 – 53 – 2k and 7 = 57 - K. Find - ã+ 7. <3i Х 5j k X>