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

Q: For which odd primes p is 3 a square mod p?

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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 by induction that if p is any real number satisfying p > -1, then (1+ p)" > 1+ np for all n e…

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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: 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: For integer n>1, nZ={nx:x in z}, nZ is set of all integer multiples of n. Prove that integer p is…

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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: For which odd primes p is 5 a square mod p?

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Q: Let p be a prime number and a be a positive integer. Prove that p | (a^p - (p-2)!a).

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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: Prove Euler’s criterion, which states that if p is an odd prime and a is a positive integer not…

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Q: For which odd primes p is 5 a square modulo p?

A: As we know that  a is Quadratic residue mod p if and only if   a(p-1) /2 =1 mod p . If p and q are…

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 an odd prime not equal to 5. Prove that either p² – 1 or p? + 1 is divisible by 10.

A: We Shall solve one question only un per the company guidelines as you have asked more than one Que.…

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

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Q: Prove that for every prime p, there exists an integer r, such that r = 16 mod p.

A: To prove: For every prime p, there exists an integer x, such that x8≡16 modp.

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: 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: 2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).

A: To prove the below statement If p is a prime number and p &gt; 5. p2−1≡0mod24

Q: Before arriving at his famous conjecture, Goldbach had conjectured that every odd positive integer…

A: Goldbach conjecture says that every integer greater than 5 is the sum of three primes.

Q: Establish that if a is an odd integer, then for any n ≥ 1 a 2 n ≡ 1 (mod 2n+2). [Hint: Proceed…

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

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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: Prove that for every ponitive integer a if aº – 6° is a prime for some positive integer 6 5 = 2 or 5…

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Q: Prove that every prime greater than 3 can be written in the form 6n + 1 or 6n + 5.

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Q: Let p be a prime number such that p> 2. Use the proof by contradiction to show that at least one of…

A: See the solution attentively I think you will understand easily.

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: prove that it is not possible to find four distinct prime numbers p, q, r, s such that ps − qr = 0.

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Q: show that If n is a positive integer and a is any integer prime to n, then a ln) = 1(mod n), where…

A: It's called Euler's generalization to Fermats theorem. I have used integer m instead of n.

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

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Q: Let p be a prime such that p mod 4 = 3. How many elements oforder 4 are in GF(pn)*?

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Q: Let p be a prime, a ∈ Z. Prove that ap − a is divisible by p.

A: Fermat's Theorem : If p is any prime and a is any integer then p divides (ap-1-1).

Q: Prove that if n is an odd positive integer, then n² = 1 (mod 8).

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Q: Prove or disprove. There is a prime of the form p = 6k – 1 such that the congruence 13x? + 7x +1 = 0…

A: We have the given Quadratic function is f(x)=13x2+7x+1 The Discriminant is given by…

Q: infinitely many primes of the form

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Q: Let p be a prime such that p = 1 (mod 4). Prove that -lF, is a square in Fp.

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Q: Let p and q be odd primes and let m and n be positive integers. Explain why U(p") O U(q") is not…

A: given, let p and q be odd primes. and n be a positive integer.

Q: Let p ∈ Z be prime. Prove that pZ is prime.

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Q: (b) Prove that if p is a prime greater than 2 then 2(p- 3)! = -1 mod p %3D

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Q: 1. It has been conjectured that there are infinitely many primes of the form n2 – 2. Exhibit five…

A: See full solution below

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: Prove that for all primes p that if p does not divide n2, then p does not divide n.

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Q: Let n be an odd positive integer. Find all perfect numbers of the form nn +1.

A: n=1,3,5,7,9,11,13,...nn+1=11+1,33+1,55+1,77+1,99+1,... Sequence is: 2, 28, 3126, 823544, 387420490,…

Q: Let r be a primitive root of the prime p with p ≡ 1 (mod 4). Show that −r is also a primitive root…

A: The given problem is to show that -r is the primitive root of p when r is a primitive root of prime…

Q: Prove that there are infinitely many primes of the form p = 4n + 3.

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

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Q: Let p ≥ 5 be a prime. Prove that p(p+1) | (p − 1)! +p+1. -

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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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