If possible, let berational. v2 a Then, there exist positive co -primes a andb such that - -

Q: 2. Investigate the definiteness of the following quadr

A: Definiteness of quadratic form: It might be positive definite, negative definite, or indeterminate…

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 p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

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

A:

Q: 3. x* + 1 is reducible over Z, for every prime p. a) True b) False

A: We have to find out whether x4+1 is reducible over Zp for every prime p or not a polynomial is said…

Q: suppose a prime p of the form 8n + 3 and that q = (p-1) / 2 is also prime. Prove that ordp (2) = 8n…

A: Consider the prime p is of the form 8n+3 and q=(p-1)/2 is also prime. Then, objective is to prove…

Q: Exercise 7.18. Suppose HG and |G| = 2p, where p is an odd prime number. Prove that either |H| = 2 or…

A:

Q: |State whether the following statements are true or false. Show working. a. For each odd prime p, a…

A:

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: If p > 2 is a prime of the form 2m +1, prove that m is a power of 2.

A: Let p be a prime and p > 2. And, p = 2m +1 Suppose 't' is an odd prime factor of m. That is, m =…

Q: x4 >is ........in Q[x] Not prime ideal ONone of the choices O Prime not maximal

A: Answer is maximal ideal , i.e , last option , and proof is given below.

Q: Find all the positive prime p such that 9p+1 is a perfect cube. Namely, such that there exists an…

A:

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: |State whether the following statements are true or false. Show working. a. For each odd prime p, a…

A: We have to state whether the statements are true or false

Q: Find all the positive primes p such that 9p+ 1 is a perfect cube. Namely, such that there exists an…

A:

Q: b. Find the remainder using Fermat's Little theorem 452 mod 11.

A:

Q: For p an odd prime in Z , Prove: If -1 is a perfect square in Z/pZ then p= 1(mod 4)

A: This can be done as follows:

Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: Suppose that p is a prime with p = 5 mod 8 . Show that the congruence px2 + 6y = 1 mod 3p has no…

A: Consider the given information.

Q: For p an odd prime in Z , Prove: If p is not prime in Z[i] then -1 is a perfect square in Z/pZ

A: Given p is an odd prime in Z.Assume that (-1) is a perfect square in Z/pZ.Then, there exists an…

Q: Let p be a prime divisor of the Fermat number Fn 22" + 1. : 2n+1 (a) Show that ord, 2 (b) Use…

A: As per the question we have the Fermat number Fn = 22ⁿ+1 and its prime divisor p. Here we will use…

Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p is not prime in Z[i]

A: Given that p is odd prime. This implies that p is not equal to 2.

Q: ) Show that if p is a prime such that m +1< p < 2m + 1, then p divides (2m+1).

A:

Q: (a) Characterize the primes p for which 63 is a quadratic residue.

A: Quadratic residue: Let p be any odd prime member and gcd(a, p)=1. Then a is said to be a quadratic…

Q: Find all positive primes p such that n = 1001p satisfies that a" = a mod n for all a.

A:

Q: DuppOse that a anа т аге intege Such that a, аre al diпerent IMod- ulo m, with am-1 = 1 mod m. Show…

A:

Q: Show that for any odd prime p, the Gauss, G(p), can be rewritten as p-1 G(p) = 5². x=0

A: To Show- Show that for any odd prime p, the Gauss, Gp, can be rewritten as Gp = ∑x=0p-1ζpx2.

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

A:

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: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

Q: Use Fermat's theorem to show that for any two distinct odd primes P.9, and any integer C: CM-c-c +c…

A:

Q: If gcd(m,p) = 1 and p is prime, show that gcd(m,pk ) = 1 for all k≥ 1.

A: To show, If gcd(m, p)=1 and p is prime, show that gcd(m, pk) = 1 for all k≥1.

Q: Let n = pq be the product of two odd primes. Recall that there are no primitive roots modulo n. Show…

A: We use the idea that if primitive root exists then x^2 = 1 mod (m) has only 2 solutions. The…

Q: Find (with proof) all positive primes p such that 217p is a Carmichael number.

A: Carmichael number - A number n is a Carmichael number if and only if n=p1p2…pj , a product of at…

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: 3. Find the residue of 20211202 mod2022 using Euler's Theorem.

A:

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: Show that in Zm there is no nilpotent element except 0 by letting m be the product of three distant…

A: Statement : In Zm, there is no nilpotent element except 0 We need to prove the  above statement by…

Q: Let m >1 be an integer, and let R = Z/mZ. Prove that R is nice if and only m is prime. Proof. ...

A: Given: m&gt;1 is an integer and R=Z/mZ. To show R is nice if and only m is prime.

Q: (i) Prove that there are infinitely many primes of the form p = 6n – 1. -

A: Given:    Infinitely many primes of the form p=6n-1     Proof:         Suppose p1, . . . , pn are…

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: Use Wilson's Theorem to prove that for primes p congruent to 1 modulo 4 the quadratic congruence 1'…

A: Given The quadratic congruence x2≡-1(mod p) To prove  x2≡-1(mod p) has a solution if p≡1(mod 4)…

Q: Prove for every pair p, q of distinct primes that

A: To prove that for every pair p, q of distinct primes that pq is irrational.

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

A: Handwriting solution.

Q: If |G| = pq, where p and q are primes that are not necessarily distinct,prove that |Z(G)| = 1 or pq.

A: Given: G=p.q where p &amp; q are primes To prove: ZG=1 or pq

Q: Determine a general congruence condition satisfied by all primes p with p = 3(mod4) such that the…

A: As per the question we are given a number theory problem. Where we have to find a congruence…

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

A: