Let a,b, and c be prime numbers that are each odd and distinct from each other....
Let a,b, and c be prime numbers that are each odd and distinct
from each other. How many solutions does the congruence = 1 mod
abc have?
Homework Answers
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Let a,b, and c be prime numbers that are each odd and distinct
from each other....
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...
76.Let p be an odd prime. Prove that if Ord, (a) = his even, then a/2...
76.Let p be an odd prime. Prove that if Ord, (a) = his even, then a/2 = -1 mod p. 77.let p be an odd prime. Prove that if Ord, (a) = 3, then 1+ a + a? = 0 mod p and Ord,(1 + a) = 6. 78.Show that 3 is a primitive root modulo 17. How many primitive roots does 17 have? Find them.
Let p be an odd prime. Prove that if g is a primitive root modulo p, then g^(p-1)/2 ≡ -1 (mod p).
Let
p be an odd prime. Prove that if g is a primitive root modulo p,
then g^(p-1)/2 ≡ -1 (mod p).
Let p be an odd prime. Prove that if g is a primitive root modulo p, then go-1)/2 =-1 (mod p) Hint: Use Lemma 2 from Chapter 28 (If p is prime and d(p 1), then cd-1 Ξ 0 (mod p) has exactly d solutions).
Let p be an odd prime. Prove that if g is a primitive...
please prove proofs and do 7.4 7.2 Theorem. Let p be a prime, and let b...
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...
Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perf...
Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;} (1) Show that if p1 (mod 3) then Tp (p 1)/3.
Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;}
(1) Show that if p1 (mod 3) then Tp (p 1)/3.
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 +...
Let p be an odd prime. Write p in the form p = 2k + 1...
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.
6. Let n be any positive integer which n = pq for distinct odd primes p....
6. Let n be any positive integer which n = pq for distinct odd primes p. q for each i, jE{p, q} Let a be an integer with gcd(n, a) 1 which a 1 (modj) Determine r such that a(n) (mod n) and prove your answer.
Problem 6: Let p be an odd prime number, so that p= 2k +1 for some...
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.
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 quadrut...
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. 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...
76.Let p be an odd prime. Prove that if Ord, (a) = his even, then a/2 = -1 mod p. 77.let p be an odd prime. Prove that if Ord, (a) = 3, then 1+ a + a? = 0 mod p and Ord,(1 + a) = 6. 78.Show that 3 is a primitive root modulo 17. How many primitive roots does 17 have? Find them.
Let
p be an odd prime. Prove that if g is a primitive root modulo p,
then g^(p-1)/2 ≡ -1 (mod p).
Let p be an odd prime. Prove that if g is a primitive root modulo p, then go-1)/2 =-1 (mod p) Hint: Use Lemma 2 from Chapter 28 (If p is prime and d(p 1), then cd-1 Ξ 0 (mod p) has exactly d solutions).
Let p be an odd prime. Prove that if g is a primitive...
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...
Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;} (1) Show that if p1 (mod 3) then Tp (p 1)/3.
Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;}
(1) Show that if p1 (mod 3) then Tp (p 1)/3.
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 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.
6. Let n be any positive integer which n = pq for distinct odd primes p. q for each i, jE{p, q} Let a be an integer with gcd(n, a) 1 which a 1 (modj) Determine r such that a(n) (mod n) and prove your answer.
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.
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...
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