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

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

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Q: Show that if p is an odd prime, then every divisor of the Mersenne number 2p − 1 is of the form 2kp…

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Q: 2. Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? +1 is divisible by 10. 3.…

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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: 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 are the prime ideals of Z2xZ2

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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: 8. Prove that if the integer n has r distinct odd prime factors, then 2" | ¢(n).

A:

Q: Suppose that the prime factorizations of two positive integers x and y are expressed using a common…

A: The prime factorization is breaking down a number into the product of only prime numbers. These…

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: Prove that p=5 and q=2 are the only pair of prime numbers such that p−q=3.

A: you can solve it by using the solution of Diophantine equation. So , Here we have to prove  p = 5…

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: 3. Prove that p is irrational for any prime p.

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the third question.  Please…

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

A:

Q: 10. Let n be an even perfect number. Prove that n=2*(2k+1 – 1), for some integer k.

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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: Find (characterize) all primes p > 3 that satisfy (금) 1.

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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: 2. Prove existence and uniqueness of a p-

A: Introduction: Two lines l and m are perpendicular if there exists a point A that lies on both l and…

Q: Prove that for all primes p2 5, p + 1 cannot be a perfect square. Recall that is a perfect square if…

A:

Q: prove that there exists a prime number p such that n <p< 2n for n = 10,11,12,13,14, 15, 16, 20.

A: for n=10 then take p=19, 10&lt;=p&lt;=20  for n=11 then take p=19, 11&lt;=p&lt;=22 for n=12 then…

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: Compute the supremum and infimum of the set A = {n/(2n+1) : n ∈ ℕ}

A: We have to find the supremum and infimum of the following set A =n(2n+1):n∈ℕ

Q: 6. (a) Show, using the pigeonhole principle, that in any set of 5 integers, at least two have the…

A: (a) Show that any set of 5 integers at least have two the same remainder when it is divided by 4…

Q: Show that if p is an odd prime, then there are exactly (p − 1)∕2 quadratic residues of p among the…

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 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: 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: 7. Prove that two consecutive integers are always relatively prime. That is, show: gcd(a, a + 1) = 1

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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: 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: Using the Law of Quadratic Reciprocity, prove that for an odd prime p ≠ 3 (3/p) = { 1 if p congruent…

A: It is given that qp=qpif p≡1mod4 or q≡1mod4=-qpif p≡1mod4 or q≡1mod4 We know that when p and q are…

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: Let n > 1 be an integer. Show that if pn for all primes p≤ √n, then n is either prime or the product…

A: Introduction: Proof by contradiction, sometimes referred to as indirect proving or the reductio ad…

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: (a) Find two consecutive primes that differ by at least 10. (b) Prove that there are arbitrarily…

A: According to the given information, for part (a) it is required to find the two find the two…

Q: List the first 3 positive prime numbers d in Z so that the quadratic integers in Q[Vd] are precisely…

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Q: Let a relatively prime positive integers. If there exist integers s and t with as + ht t - 11 then

A: if gcd (a,b) =d then there exists integers m and n such that d = a m + b  n  and d is the smallest…

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

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Q: is.in Q[x] Not prime ideal O Prime not maximal Maximal None of the choices

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Q: Prove that If p is prime, p divides k^2 + m^2 and p divides m^2 + x^2, then p divides k^2 + x^2

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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: Take non-zero positive integers m and n. Show that Z/mZx Z/nZZ/mnZ if and only if m and n are…

A: Given That:To take non-zero integers m and n .   ℤmℤ × ℤnℤ ≅ ℤmnℤ if m and n are relatively prime ,…