Prove that there exists a positive integer n >1 such that 2n2 + 3n + 11 is prime.

Q: Prove Liebeck’s triplet prime conjecture: the only triplet of primes of the form p, p+2, p+4 is…

A: Here we give two different proofs for conjecture:

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

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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: (12) Let p be a prime integer, p≥ 5. Prove that p² - 1 is divisible by 24.

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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: Prove or disprove For every integer m, m(m+1) is even

A: To prove or disprove the given statement- For every integer m , m(m+1) is even .

Q: k+4 n(3n +7) k(k +1)(k + 2) 2(n+1)(n+2) k=1

A: Given ∑k=1nk+4k(k+1)(k+2) To show  ∑k=1nk+4k(k+1)(k+2)=n(3n+7)2(n+1)(n+2)

Q: Prove that every positive integer n>1 has a prime factorization.

A: Prime factorization is a way of expressing a number as a product of its prime factors.

Q: 1.) Prove every integer n >= 1, 1+5+9...+(4n-3) = n(2n-1).

A: 1.Apply mathematical induction to prove the statement. Let the given statement be Pn:…

Q: Prove that if n is an even integer greater than 2, then 2 - 1 is a composite.

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Q: Prove that if a and b are two distinct positive primes, then the number k = ab cannot be abundant.

A: To prove that if ( a ) and ( b ) are two distinct positive primes, then the number ( k = ab ) cannot…

Q: e induction to prove that 3 ≥ 7n for all integers n with n ≥ 3.

A: By using induction on n we will prove the result.So , basic step : n=3 we have…

Q: Prove that 25|(6" – 5n – 1) for all positive integers n. -

A: We can prove given statement by principle of mathematical induction: Alogrithm: Obtain P(n) and…

Q: Use induction to prove that 3 | (10n+1 + 10n + 1) for all positive integers n

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Q: prove that n(13n²-1) is divisible by 6 for every n ≥1

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Q: Let's continue simulating the process of constructing a matrix for a 3D transformation that rotates…

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Q: Let k be a positive integer. Show that 1" + 2 + 3™ +....+(n – 1)m + n™is O (n™+1). -

A: According to the given information, it is required to show that:

Q: Use induction to prove that 3 | (2²n — 1) for all nonnegative integers n.

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Q: Use mathematical induction to prove that for each nonnegative odd integer n: 2n+1 2 + 1)(n – 1) 24 |…

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Q: Let a and b be natural numbers, where a,b >= 2, such that gcd(a,b)=1 and ab=(x+y)n for some natural…

A: Given that a and b are natural numbers and  a , b ≥ 2 .  Also, given that gcda,b = 1 and ab = x+yn…

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

A: We have to prove that and  are relatively prime for every integer .We know thatThe greatest common…

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: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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Q: Show that for every positive

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Q: Prove that the integers n, n+2, n+4 are all primes if and only if n=3

A: Statement: Prove that the integers n, n+2, n+4 are all primes if and only if n=3 Proof by…

Q: Euler's formula for generating amicable pairs of integers is the following rule: = Suppose n> m > 1…

A: Note : In the Euler's formula for generating amicable pairs it is r = 2n+m (2n-m + 1)2 - 1 and not…

Q: Prove using Mathematical Induction that: The sum of the squares of the first n positive integers is…

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Q: Prove that: 3n > 2n + n2 for all integers n is greater than or equal to 2

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Q: Use Mathematical Induction to Prove that 2 + 4 + 8 + 2 = 2(2" - 1)is true for any positive integer…

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Q: Prove that there exists a positive integer n > 2 such that 4n+1 is a prime number.

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Q: Prove that n!+1 and (n+1)!+1 integer n. relatively prime for every positive

A: Given, n!+1 and n+1!+1are two numbers and to prove that they are relatively prime.

Q: Use any methods of proof to verify that n 1² + 2² + 3² + ... + n² = 1² (n · (n + 1)(2n + 1) 6

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Q: induction to prove that 12 + 22 + ... + n2= n(n+1)(2n+1)/6  for all n>0.

A: To prove by induction :Let the proposition P(n): 12 + 22 + ... + n2 = n(n+1)(2n+1)/6Here n is…

Q: Prove that there is some positive integer n such that n, n + 1,n + 2,... , n + 200 are all…

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Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…