3. Euler made the following conjecture: Let a be any natural mumber and p, q primes such that p= ±q mod 4a. Then a is a square mod p if and only of a is a square mod q. Use this conjecture to give a proof of Law of Quadratic Reciprocity.

Q: The linear congruence in two variables ax + by ≡ c (mod m), where a, b, c, and m are integers, m > 0…

A: As we know-the congruence ax+by≡c (mod m)has dm incongruent solution if   d=(a,b,c)|c& no…

Q: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

Q: For congruences of primes p = 1 (mod 4), a. If a is a quadratic residue modulo p, and p = 1 (mod 4),…

A:

Q: 12. Prove that if gcd(a, n) { b, then ax = b (mod n) has no solutions.

A: To prove the given statement, it is equivalent to prove its contrapositive statement which is "if…

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

A:

Q: Use the Chinese remainder theorem to show that an integer a, with 0 ≤ a < m = m1m2⋯mn, where the…

A:

Q: Suppose m, n e Z† are relatively prime. Prove that for all a, b e Z, a = b (mod mn) iff a = b (mod…

A: The congruence relation a≡b mod n implies that a-b is a multiple of n. That is n| a-b or a=kn+b If…

Q: Show that: Ifp is odd prime of the form 4k+1, then find a solution of x +1 0 mod p

A: We shall use Euler’s Criterion here. Euler’s criterion states as follows: “Let p be a prime. The…

Q: a) How can an inverse of a modulo m be used to solve the congruence ax ≡ b (mod m) when gcd(a,m) =…

A:

Q: (5) Show that if n ɛ Z+ is odd which is divisible by k distinct primes, then the congruence x2 = 1…

A: Chinese Remainder Theorem: If a1,a2,...,ak are integers, and m1,m2,...,mk are pairwise relatively…

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: The linear congruence 7x = 3(mod16) is solvable of solution: O None of the choices

A: The given problem is to solve the linear congreunce equation, we have to use congruence 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: 2. Let a be a positive integer. Explain why the linear congruence az b (mod a + 1) has a solution in…

A:

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: 4. The least positive residue of 2^1000000 mod 17 is: * 1 None of these

A: Question 01: By Division algorithm,  10,00,000= 16*62,500 and since 17 is prime and 17 is not…

Q: Suppose that p is a prime such that p = 5 (mod 8). Let k E Z such that p = 8k+5. For any a E Zp,…

A: 1. Let a∈ℝn. Then a is called quadratic residue modulo n if there exists an integer x such that…

Q: Suppose m,n e Z+ are relatively prime. Prove that for all a,b e Z, a = b (mod mn) iff a = b (mod m)…

A: Given: a,b∈Z and  m,n∈Z+ are relatively prime .

Q: Find all solutions to the following quadratic congruences by first making the leading coefficient 1…

A: Topic:- algebra

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…

A:

Q: 4. Exercise $2.5 #58a. Let a and b be integers not divisible by the prime mumber p. (a) If aP = mod…

A: Given the data

Q: Use Hensel's Lemma to find a solution to the following congruence: a +3 x + 3 = 0 mod 54.

A: Let f(x) be a polynomial , p is a prime number . If x=a is a solution of congruence equation…

Q: 3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6). (a) Check this…

A:

Q: 1. Suppose m, nEz* are relatively prime. Prove that for all a, bEZ, a =b (mod mn) if a =b (mod m)…

A: Note that: a is congruent to b modulo m if and only if m divides (a-b).

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: #5. Find all incongruent solutions of each quadratic congruence below. (a) x? = 23 mod 77 (Hint:…

A:

Q: Prove that for any prime, p = 1 (mod 4) there are exactly (p-1)/4 incongruent values of a 0 (mod p)…

A: By theorem (a,p)=1 then xn≡a modp has g.c.d(n,p-1) solution or no solution according as…

Q: Use the construction in the proof of the Chinese remainder theorem and backward substitution to find…

A: Given

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

A:

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A:

Q: (a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b)…

A: (a) Given that p is an odd prime.  The quadratic formula mod p is given by: ax2+bx+c≡0mod p Here, p…

Q: Let a, b, c, and m be positive integers. (a) Prove that the linear congruence ax = c (mod m) has…

A: Solution: a) Consider the congruence  ax≡cmod m---(1) Let, the congruence equation (1) has a…

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

A:

Q: 1. Use Euler's Theorem to prove Q265 = a for all a E Z. a (mod 105)

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: You and your friend Lara are considering the linear congruence ax = b (mod m), where m = 6 x 11 x…

A:

Q: Show that if p is an odd prime, then −1 is a quadratic residue of p if p ≡ 1 (mod 4), and −1 is not…

A:

Q: Show that if p is an odd prime and a is an integer not divisible by p, then the congruence x2 ≡ a…

A: According to the given information, Let p is an odd prime and a is an integer not divisible by p…

Q: Prove that if a > 0, k > 1, and ak -1 is prime, then a = 2 and k is prime. %3D

A:

Q: 4n, then 9| (unt1 ++1 e, establish the following: (a) 3 = Un (mod 2), hence u3, U6, Ug, ... are all…

A:

Q: 5 is a quadratic non- residue mod 17

A: The quadratic residues mod17 are { 1,2,4,8,9,13,15,16}

Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

A:

Q: 12. Prove that if gcd(a, n) b, then ax = b (mod n) has no solutions.

A:

Q: Let be an odd prime. Prove that for all i with 1 sisp-2 one has p| 1° + 2' + ... + (p – 1)'.

A: Given : p is an odd prime To Prove :          p| 1i+2i+. . .+p-1i for all i with 1≤i≤p-2

Q: Show that for x and y integers, and m a positive integer x = y mod m implies x = y mod m.

A: This is a problem of Number Theory, Modular Arithmatic.

Q: Solve the following system of congruence's by using Chinese Remainder Theorem x = 1 (mod 3) x = 2…

A:

Q: Find all solutions to the congruence 10x≡5 mod 15 (b) Prove that 7 divides 32n+1+ 2n+2for any…

A: In  (a) we use the extended Euclidean algorithm to solve the problem. In (b) we use the mathematical…

Q: 2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35). [Hint: From Fermat's theorem a = 1 (mod 7) and…

A: Fermat's Little Theorem:- It states that if p is any prime and a is any integer then, ap≡amodp…

Q: Suppose a, b ∈ Zare relatively prime, and c|a. Prove gcd(b, c) = 1.

A: Handwriting solution.

Q: Prove that if p is a prime in Z that can be written in the form a2 + b2,then a + bi is irreducible…

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Q: The linear congruence 8x = 16(mod14) is solvable. False True

A: