Describe congruence conditions on p > 2 an odd prime that classify exactly when 11 is a quadratic residue mod p. (Hint: the answer consists of 10 congruence classes mod 44.) Exercise 2

Q: Prove that there are infinitely many primes congruent to 3 mod 8. previous part.

A: We have to show that there are infinitely many primes congruent to 3 mod 8. we will proof this by…

Q: Let p be a prime. Show that -2 is a quadratic residue modulo p if and only if p = 1 mod 8 or p = 3…

A: Definition(Quadratic Residue Modulo): In number theory an integer qis called a quadratic residue…

Q: ​Q3// Use Chinese theorem to solve the following congruence: X ≡ 2 mod 3 X ≡ 1 mod 5 X ≡ 2 mod 4

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Q: Q// Let m = 5, 1- Write the Addition table (Cayley's Table). 2- Write the congruence classes of…

A: For m=5 , our set is Z5 .

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A: In congurent modulo n, ax≡bmod n has exactly 'd' solutions if 'd' divide 'b'where d=gcda,n otherwise…

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Q: ve squares modulo p) We can use quadratic reciprocity to answer tr question: "How many residue class…

A: Given:- ∑x=0p=1xp=0 and if that x does not equal to zero (mod p)…

Q: Exercise 4.21. Prove that x2 = 1 (mod p) has exactly two solutions when p> 3 is a prime. 70

A: We shall solve it using basic knowledge from elementary number theory, in particular the properties…

Q: Let p be a prime and a a quadratic residue of p. Show that if p =1 (mod 4), ais also a quadratic…

A: Let P be an odd prime and Let a be a quadratic residue mod p ∵ P-a≡-a (mod p) ∴(P-a)p-12≡(-a)p-12…

Q: 5. Let p be an odd prime, and suppose that ged(a, p) = 1. Show that the congruence z = a (mod p*)…

A: p is an odd prime and gcd(a,p)=1We have to show the congruence x2=a mod pk has exactly 1+ap…

Q: Find all values of that are solutions to the following system of congruences (Show your work!): x =…

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Q: Let p be a prime number. The polynomial p(x) = x²+x+1 is irreducible over Zp if and and only if…

A: Given that for a prime number p, the polynomial px = x2+x+1 is irreducible over ℤp if and only if…

Q: We'll now try to prove the following result in 2 steps: Theorem: Suppose that p is an odd prime. Let…

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Q: Consider the product of the nonzero quadratic residues modulo p. For example, modulo 5, the nonzero…

A: We consider the product of the nonzero quadratic residues modulo p. For example, modulo 5, the…

Q: Suppose that p≥ 5 is prime. Show that -3 is a quadratic residue (mo p) if p = 1 or 7 (mod 12) and a…

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Q: Suppose that residue(modp) 8) Let odd prime P. atzis Show that has be an not Diophantine…

A: Given that a is not a quadratic residue mod p. That is there does not exists an integer z such that,…

Q: If m>4 is a composite integer, then O (m-1)!=1 mod m (m-1)!=2 mod m (m-1)!=0 mod m None of these

A: Since you have asked multiple question, I solved first one for you. If you need solution of other…

Q: Make sure that all your answers are explained carefully. (a) Is 2 a quadratic residue modulo 13?…

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Q: Let p be an odd prime and let a be a quadratic residue modulo p. Prove that -a is also a quadratic…

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Q: 2) i) Show that 5 is a primitive root modulo 17. Find the all integers, less than 17, whose orders…

A: To show that 5 is a primitive root modulo 17, we need to demonstrate that it has the property of…

Q: 3. Exercise 55.2 #12. Let p be an od«

A: Given, p be an odd prime number.

Q: 1. Solve the congruence x² + 1 = 0 mod 13³.

A: To solve the congruence  , we can use the fact that  is a prime number.The congruence  implies that…

Q: (9) Let a, b, and m be integers with m 21 and (a,m) = 1. Prove that the congruence ax=b (mod m) has…

A: a, b and m are integers with m ≥ 1 and (a, m) = 1.We will prove that ax ≡ b  (mod m) has a unique…

Q: Prove that 13n13 + 11n" is always divisible by 24.

A: Consider the given expression.

Q: (a) Use cases based on congruence modulo 3 and properties of congruence to prove that for each…

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Q: b) Determine whether or not x? = 85 (mod 97) has a solution.

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Q: a. Find a reduced residue system modulo 15. b. Find the multiplicative inverse of each residue.

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Q: We want to test the primality of 105 by using the Rabin-Miller test, with the basis 2. Describe and…

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: 2 + v - , 2+z+ 11. 0. di

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Q: Find a and b that make the following statement true, and prove your answer. Let p be an odd prime.…

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Q: Let p be a prime of er, 1 < a <p-1. 7 Hulo p.

A: According to the question, Let's start by clarifying some terms A quadratic residue is an integer…

Q: Let a be a quadratic residue of the odd prime p. (a) Show that a is not a primitive root modulo p.…

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Q: Suppose p is a prime congruent to 3 modulo 4. (a) Prove that the congruence r = -1 (mod p) has no…

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Q: 11. In 1970 the Department of Natural Resources released 1000 splake (a crossbreed ofE fish) into a…

A: The Malthusian growth model is P(t) = P0ert, where P0 is the initial population.