Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p, at least one of p,p+2,p+4 is composite.

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: . Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when…

A: Consider an integer x that has a remainder of 2 when it is divided by 5 Thus x=2 mod 5 0r x=5k+2 It…

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: Let n be a perfect number greater than 6. Prove that n = m²p for some positive integer m and odd…

A: Consider the given that n be the perfect number greater than 6.We have to prove that n=m2p ,where m…

Q: Show that if n is an even perfect number, then n = 2P-'Mp for some Mersenne prime Mp 2P – 1.

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Q: Prove there are infinitely many prime numbers

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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: 24. Show that if p is a prime and k is an integer such that 1sksp-1. then p divides ().

A: 24. we have to show  that for a prime p with r≤k≤p-1      p divides  pk  Notation: pk=p!p-k! k!

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: Show that there are prime numbers p and q such that p+q=128.

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Q: Number theory.

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Q: 1. Prove that the Mersenne number M13 is a prime; hence, the integer n = 2' (213 – 1) is perfect.…

A: handwriting solution

Q: Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

A: Consider the given statement as follows. Show that 2p-3!+1≡0mod p, for all odd primes p. Consider an…

Q: Show that for every positive

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Q: How do I prove: If n > 1 is a natural number then there is a prime number p such that p|n.

A: This is a theorem of real analysis in which we have to find that there is a p who divides n.

Q: 1) Prove that the numbers -197, -239 are prime numbers 2) Find all prime numbers between -90 and 90.

A: number system

Q: 21 divides 4n+1 + 52n-1

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

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Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

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Q: Show that there is a prime number p such that p+4 and p+6 are also prime numbers.

A: Prime number is a number p having only two divisors p and 1 only.

Q: Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p ≥ 5, at least one of…

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Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. In a proof by…

A: We have for each positive number n P(n)=5n-1 To prove P(n) is divisible by 4 using mathematical…

Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 <n < 20. Circle any numbers that are…

A: Composite numbers contain prime factors other than 1. Such as if a number has prime factors of 2, 3,…

Q: 9. Show that if p is an odd prime and p = 3 mod 4 then p is not the sum of the squares of two…

A: In the given question we have to prove that if p is an odd prime and…

Q: 3. The GCF of any two distinct prime numbers is 1. (Distinct they are not the same number) means

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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: 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…

Q: how many pairs of primes p and q are there such that q-p=3? Prove your answer.

A: Here, we need to find all the possible pairs of prime numbers p and q such that the difference (q-p)…

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

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Q: 18. Numerical evidence makes it plausible that there are infinitely many primes p such that p+50 is…

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Q: Let p1, p2, . . . , pn be primes. Show that p1 p2.....pn +1 is divisibleby none of these primes

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Q: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.

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 2n−1 is prime then n is prime.

A: We have to show that if 2n-1 is prime then n is prime.

Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

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: Prove that any integer greater than or equal to 48 can be represented as the sum of nonnegative…

A: Principle of mathematical induction: For each and every natural number n, mathematical induction is…

Q: 2. If p > q > 5 and P and q are both primes, prove that 24 p - q² |

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Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

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Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 <n < 20. Circle any numbers that are…

A: We know that if a number k is divisible by m   Then  There exists a natural number n such that k=nm

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 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: Find all numbers p such that all six numbers p, p+2, p+6, p+8, p+12 and p+14 are primes.

A: For the given statement

Q: If n and a are prime to 7 then prove that nº – a® divisible by 7?

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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: 74. P is a natural number of at least 6 digits and its left- most digit is 7. When this leftmost…

A: Introductions :

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

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Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 < n < 20. Circle any numbers that…

A: Table of values for 10n-1 and 10n+1 for n between 1 and 20. n 10n-1 10n+1 1 9 11…

Q: Let p2 q be primes greater or equal to 5. Show disprove that 24 divides p² – q². or

A: Consider the provided question,

Q: is a prime number and that n is an integer such that p|n^2 prove that it follows p|n

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