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ame: . (10 points) Let p > 3 be any prime number. (a) Show that p...
Write your own answers 10. Let p be a prime number, and let a be an...
Write your own answers
10. Let p be a prime number, and let a be an integer that is not divisible by p. Prove that the congruence equation ax = 1 mod p has a solution X e Z.
Question 1 2(a) Let m>1 be an odd natural number. Prove that 13-5.-(m-2) (- 2-4-6. (-1)...
Question 1 2(a) Let m>1 be an odd natural number. Prove that 13-5.-(m-2) (- 2-4-6. (-1) (mod m) (m-1) (mod m [Hint : 1 i-(m-1 ) (mod m), 3 Ξ-(m-3) (mod ") , . .. , m-2 1-2 (mod m)] 14 (b) If p is an odd prime, prove that Hint: Use Part (a), and rearrange the Wilson's Theorem formula in two different ways
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.
Let J be the ring of integers, p a prime number, and () the ideal of...
Let J be the ring of integers, p a prime number, and () the ideal of J consisting of all multiples of p. Prove (a) J@) is isomorphic to Jp, the ring of integers mod p. (b) Using Theorem 3.5.1 and part (a) of this problem, that J, is a field.
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.
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...
this is number theory i need help with thanks alsonlls show all work Assume a, b,...are...
this is number theory i need help with thanks alsonlls show
all work
Assume a, b,...are integers, r, s, t > 1, m > 2, p =prime> 2. 1. Write c= (m) and let 91, 92,...,q* be all the distinct prime factors of c. Suppose that (a,m) = 1 and ac/4 # 1(mod m), 1sisk. Prove that a is a primitive root (mod m). Prove that 2 is a primitive root (mod 11). 3. Find the indices of 3, 4...
Let p be a prime number. Prove that 19–1 + 2P-1 + ....(p – 1)p-1 =...
Let p be a prime number. Prove that 19–1 + 2P-1 + ....(p – 1)p-1 = -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 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...
4. (a) [3] Let p be prime and let M, denote the number 2P – 1....
4. (a) [3] Let p be prime and let M, denote the number 2P – 1. The number M, is called a Mersenne number, and if it is prime, it is called a Mersenne prime. There is a test, called the Lucas-Lehmer Test, that gives a necessary and sufficient condition for My to be prime. It is always used to verify that a Mersenne number, suspected of being prime, is indeed a Mersenne prime. Give the statement of this test....
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10. Let p be a prime number, and let a be an integer that is not divisible by p. Prove that the congruence equation ax = 1 mod p has a solution X e Z.
Question 1 2(a) Let m>1 be an odd natural number. Prove that 13-5.-(m-2) (- 2-4-6. (-1) (mod m) (m-1) (mod m [Hint : 1 i-(m-1 ) (mod m), 3 Ξ-(m-3) (mod ") , . .. , m-2 1-2 (mod m)] 14 (b) If p is an odd prime, prove that Hint: Use Part (a), and rearrange the Wilson's Theorem formula in two different ways
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.
Let J be the ring of integers, p a prime number, and () the ideal of J consisting of all multiples of p. Prove (a) J@) is isomorphic to Jp, the ring of integers mod p. (b) Using Theorem 3.5.1 and part (a) of this problem, that J, is a field.
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.
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...
this is number theory i need help with thanks alsonlls show
all work
Assume a, b,...are integers, r, s, t > 1, m > 2, p =prime> 2. 1. Write c= (m) and let 91, 92,...,q* be all the distinct prime factors of c. Suppose that (a,m) = 1 and ac/4 # 1(mod m), 1sisk. Prove that a is a primitive root (mod m). Prove that 2 is a primitive root (mod 11). 3. Find the indices of 3, 4...
Let p be a prime number. Prove that 19–1 + 2P-1 + ....(p – 1)p-1 = -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...
4. (a) [3] Let p be prime and let M, denote the number 2P – 1. The number M, is called a Mersenne number, and if it is prime, it is called a Mersenne prime. There is a test, called the Lucas-Lehmer Test, that gives a necessary and sufficient condition for My to be prime. It is always used to verify that a Mersenne number, suspected of being prime, is indeed a Mersenne prime. Give the statement of this test....
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