Suppose p is a prime such that p ≡ 1 (mod 4). Show that the sum of the quadratic residues mod p lying in {1, . . . , p − 1} is given by (p(p−1)/4). Hint: Pair each quadratic residue awith its additive inverse −a.

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: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.

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Q: Show that log, q is irrational for any distinct prime numbers p and q. (Distinct here p# q.) means

A: To Determine :-  logp q is irrational for any distinct prime numbers p and q.

Q: If the A.M. and G.M. between two numbers are in the ratio mn, then prove that the numbers are in the…

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Q: Suppose p is a prime such that p ≡ 1 (mod 4). Show that the sum of the quadratic residues mod p…

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Q: Consider the sentence (3xP(x) ^ 3xQ(x)) 3x(P(x) ^ Q(x)). Which of the following is true? A. The…

A: Let P(x) : x is such that x ≡ 1 (mod 3)         Q(x) : x is such that x ≡ 2 (mod 6)    Then,…

Q: Show that x2 = Y mod p has

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Q: 2. Let a be a positive integer. Explain why the linear congruence ax = b (mod a + 1) has a solution…

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Q: Let p be a prime that does not divide n. Prove that Φpn(x) =Φn(x p)/Φn(x).

A: Given: p be a prime number that does not divide n. To Prove: Φpnx=Φnxp/Φnx.

Q: 1. Prove (2p+1) = 2 (mod p) when p is any prime.

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Q: Let a and b be positive integers and let b= qa +r, where 0 ≤r <a. Prove that (a) 2b - 1 = 2" - 1…

A: Let  and  be two positive integers.Let  where (a) We  have to prove that      Now, observe that…

Q: Let x E Z. Prove that either x2 = 0 (mod 4) or x2 = 1 (mod 4).

A: By the help of even and odd integer's we prove the result. The answer is given as follows :

Q: a) Let p be an odd prime. Show that a = -1 mod p has a solution if and only if p = 1 mod 8.

A: Consider the provided question,Hello. Since your question has multiple parts, we will solve first…

Q: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

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Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

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Q: Let m be the number of these remainders which exceed p/2. Then ( #) = (-1)m.

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Q: 2.23 Find all four solutions to the equation x²10 (mod 35).

A: The given equation is: x2-1≡0 mod 35 We have to find all four solutions to the given equation.

Q: Theorem 1. Let r, y, z be positive integers where r is odd and y is even. Then r, y, z is a…

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

Q: Work the modular arithmetic problem. (11.10)(mod 7). (11 10) (mod 7) = (mod 7)

A: (a)   (11.10)mod 7  =  110 mod 7                            The remainder when 110 is…

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: 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: 1. Compute the least nonnegative residue of 10k + 1 modulo 13 for k 1, 2, 3, 4.

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Q: Let be a rational prime, with p = 1(mod 4). Prove that p is the product of two Gaussian primes that…

A: We will use the basic knowledge of number theory to answer this question properly and completely.

Q: Let f₁=1,₂ = 1 and for all neN,n23,f=f+f2. Prove 1+√5 1-√√5 2 2 that for all neN, S √5

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Q: 1.34. This exercise begins the study of squares and square roots modulo p. (a) Let p be an odd prime…

A: Given:(a) Let be an odd prime number and let be an integer with does not divide . (b) (i) (ii)…

Q: Prove or disprove the following proposition: Let the Jacobi symbol: () 1. %3D Then a is quadratic…

A: a is said to be a quadratic residue modulo n if x2=a(mod n) is solvable for some x.  Otherwise, it…

Q: Prove by Contradiction: There are no integers x and y such that = 4y+2.

A: We have to prove the statement "There are no integers x and y such that x2=4y+2" by contradiction.…

Q: Find all the non-congruent solutions to  4x ≡ 6(mod 21)  express solutions using the stand…

A: Introduction: A linear congruence is generalized as ax≡bmod n. The greatest common divisor of a and…

Q: Prove that for any natural number n, 2(√√n +1 − 1) <1+ 1 √2 + <2√n.

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Q: Suppose that p and q = 4p+ 25 are both prime. Show that p is a quadratic residue mod q.

A: Given p and q are both prime. and q=4p+25 To show that p is a quadratic residue mod q.

Q: Recall the Fibonacci numbers Fn are defined by Fo = 0 , F1 = 1 and Fn Fn-1 + Fn-2 for all n > 2 .…

A: Consider the provided question, Fibonacci numbers Fn are defined by F0=0, F1=1 and Fn= Fn-1+Fn-2 for…

Q: Prove that for any odd prime p different from 3, ) = 1 if and only if p = ±1 (mod 12). (hint: Use…

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Q: 11.5. For each part, find an x that solves the given simultaneous congruences. (a) x = 3 (mod 7) and…

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Q: For any integer a, prove that a = 0 or 1 (mod 3).

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Q: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: y if p-a is a

A: According to the question, This theorem is a special case of properties regarding quadratic residues…

Q: Suppose that p is a prime number and g is primitive root. Prove that the least residues of 4 9²,9ª,…

A: Step 1:To prove that the least residues of g2,g4,g6,gk,...,gn−1 are all quadratic residues modulo n,…

Q: Theorem 5.3. Let a be an odd integer. If a3 we have (29) a 2 = 1 (mod 2º), so there are no primitive…

A: Proof:                 We shall prove by induction on α. First we consider α=3. Let α be an odd…

Q: Use Fermat's theorem to Reduce 50250 (mod 83) to a number in the range {0,1

A: Here we will find out the required value.

Q: (c) Find all integers of order 6 mod 13 and all primitive roots mod 13. Show your working in both…

A: This is a question of Number Theory.

Q: 18.) Compute fg (Note: each should only be one In 2₁, we write -1 as f(x)=x²-3x +2, term in the…

A: fg(x) = f(x)g(x)

Q: 8.5. Find all incongruent solutions to each of the following linear congruences. (a) 8x = 6 (mod 14)

A: