Show that there is only one prime p such that p² + 20 is also prime.

Q: Let m be the least common multiple of a and b. Prove that if p is a prime divisor of m, then either…

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Q: There are infinitely many primes of the form 6n+5.

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Q: Show that C(p,1), C(p,2),..., C(p,p-1) are all divisible by p if and only if p is a prime.

A: As per the question we have to prove that C(p,1), C(p,2),..., C(p,p-1) are all divisible by p if and…

Q: An integer M is called B-power-smooth if every prime power pe dividing M satisfies pe ≤ B. For…

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Q: Prove that if p and q are distinct primes, then pq | (p9-1+qp-1-1).

A: p and q are distinct prime. We have to show that pq|pq-1+qp-1-1

Q: Let p be a prime larger than 3. 1. Show that the remainder, when p is divided by 6, is 1 or 5. Note:…

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Q: > 2 is a prime, then (12) Prove that if p and (1 2... p-1 p) generate Sp.

A: Solution:                  Let H=1,2,  1,2,3.....n  and H be the smallest subgroupcontaing them.Then…

Q: Prove that if a > 1, then there is a prime p so that a < p< a! + 1.

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Q: Show that for every positive

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Q: 2. Make a conjecture about the sum of primes based on the following pattern: 4= 2+2 6=3+3 8=3+5…

A: Given:4=2+26=3+38=3+510=3+712=5+7and so on...

Q: An integer n is divisible by k e Z when there exists another integer m satisfying n = Integers that…

A: Given is a definition of a factor and a perfect number.   To Prove: (a) At least one perfect number…

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

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Q: Suppose that p is a prime. Prove that p | α if and only if p | α2.

A: Since, If p|α , this is obvious that p|α2.

Q: Suppose that a, b are two integers such that 3a? – b3 = 1. Does it follow that a and b are…

A: Given that a, b are two integers such that 3a2-b3=1.

Q: Let p and q be primes, Prove that plq if and only if p = = 9.

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Q: Prove that every prime p ≥ 5 is of the form p = √(24k + 1 )for some integer k.

A: To Prove that every prime p ≥ 5 is of the form p =√(24k + 1 )for some integer k. Or equivalently, p2…

Q: Consider the following statement: For all integers a and b, if a is even and b is a multiple of 6,…

A: The statement and the proof are already given, we just need to arrange the steps in an appropriate…

Q: Let p be a prime number with p>=5. Prove that p must either be one more or one less than a multiple…

A: We use a fact that any number greater that 5 must be written as one of the form…