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

Q: Prove that pqr is a deficient number, where p, q, r are three distinct primes with p <q < r

A: We have to prove that pqr is deficient number where p, q and r dustinct primes  Such that p &lt;  q…

Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: 3n+2 and 5n+3 are relatively prime for every integer n.

Q: 9. Show that there are infinitely many primes of the form 16k + 1. More generally, show that for any…

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Q: S@ Find the coefficient f e in the expansion of (1-4x)2 for all non-negative integers n 6 Using…

A: Given that, (a) The expansion of 1-4x-12. Calculate the coefficients of xn. (b) Prove that…

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: Is it true that (3 x EN) (x²-3x +20 is a prime)? O True False

A: Is it true that ( there exists x in N)(x^2-3x+20  is a prime ) ?   True False

Q: Let p be a prime divisor of the Fermat number Fn= 22^n+ 1 i. Show that ordp2 = 2n+ 1 ii. Use…

A: Given p be a prime divisor of the number Fn=22n+1. (i) Since p divides Fn=22n+1, therefore: Fn ≅0…

Q: For each nonnegative integer i, what is the least residue modulo 9 or 10i?

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Q: Prove, that if 2m + 1 is a prime number, then m must be a power of 2 (m = 2n, for some n ∈ N). Prime…

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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 &gt; 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 &gt; 2. And, p = 2m +1 Suppose 't' is an odd prime factor of m. That is, m =…

Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

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Q: Let n∈Z.  Prove that x is odd if and only if 5x + 4 is odd.  Identify the type of the above…

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

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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: 111111 is prime. Prove that, in any radix b, The number 11111 (11...1), can be prime only if the…

A: As per the question we have to prove that the number 111...1 containing n number of 1's will be…

Q: .If p 5 is an odd prime, prove that either p? – 1 or p² + 1 is divisible by 10.

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Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

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: 3. Find integers s and t so that 1 7s + 11t. Show that they are not unique.

A: The division algorithm states that for any integer, a, and any positive integer, b, there exists…

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 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: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

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: Use Induction to prove that 22n+1 + 32n+1 is divisible by 5 for all n > 0.

A:

Q: If a and b are relatively prime then alb

A: Given below the detailed solution

Q: (b) Prove that there are infinitely many prime numbers of the form 6n + 5

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Q: Number theory

A: Consider the provided question,

Q: Q1. Let x and y be two integers. Prove that for all integers m 2 1, x™ – y™ is a multiple of x-y…

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: Show that 5n + 3 and 7n + 4 are relatively prime for all n.

A:

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

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

A:

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: Let p and q be odd primes. Prove that (p^2) q is not a perfect number. [Hint: Factor 1 + p + q + pq…

A: Let σp2q=σp2σq Then we have  σp2q=σp2σq=1+p+p21+q=p3-1p-1q2-1q-1≤p3p-1q2q-1

Q: If |G = 30 and |Z(G)| that |G| 5, what is the structure of G/Z(G)? What is the structure of G/Z(G)…

A: Solution:-

Q: Show that Zn has a nonzero nilpotent element if and only if n is divisibleby the square of some…

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: Show that if 2m +1 is an odd prime, then m 2" for some nonnegative integer n.

A: We solve this question by contradiction

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: Does x = a (mod p) always have a solution for every value of a, whenever p is prime?

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Q: Exercise 1. Show that there are infinitely many primes of the form 4n +3. Hint: you might find it…

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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: (This exercise constructs another proof of the infinitude of pimes) Show that the integer Qn = n! +…

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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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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.