Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the right-hand side, show that n* – 6n3 – 18n² + 6n + 1 = (n² – 3n – 1)² – 25n². b) Hence find all integersn such that n* – 6n³ – 18n² + 6n + 1 is prime. (regarding negative prime numbers also belong to primes)

Q: 5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: ve that the only perfect number of the form n = p²q where p and q are distinct primes is 28.

A:

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: 8. Prove that n!+1 and (n+1)!+1 are integer n. relatively prime for every positive

A: Given, and are two numbers and to prove that they are relatively prime.

Q: If 2n +1 is prime for some n ≥ 1, prove that n is a power of 2.(Primes of the form 2n +1 are called…

A: Given: 2n+1 is prime for some n≥1. To Prove: n is a power of 2.

Q: True or false: If n is a positive integer with d(n) = 6, then the only possible prime power…

A: n is a positive integer with dn=6.

Q: (g) 111... 111 (126 ones) (2) Prove that the numbers given in each part cannot be prime…

A:

Q: Prove by mathematical induction that for all positive integer values of n: n a 2 (r3 – r) = }(n – 1)…

A: As per our company policy we are supposed to answer only the first question. Kindly re Post other…

Q: 6. Prove that n, n+2, n +4 are all primes if and only if n = 3.

A:

Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

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: (7) The difference of two odd integers x and y is 4. Prove that x and y are relatively prime.

A: Given difference of two odd integers is 4.

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: (g) 111... 111 (126 ones) (2) Prove that the numbers given in each part cannot be prime…

A:

Q: For every integer n > 0, 1+2+22 + · + 2" = 2"+1 – 1.

A: Given: The statement 1+2+22+⋯+2n=2n+1-1 Proof: Base step: Show the statement is true when n=0.…

Q: Show that if p = q + 4a (p and q are odd primes), then (p/q) = (a/q).

A:

Q: Show that 32n 1 is always a multiple of 8 for all n = N = {1, 2, 3,...}. We say one integer A is a…

A: Step 1:Step 2:Shows a proof by induction that 32n - 1 is always a multiple of 8 for all natural…

Q: (3n)! is an integer for all n > 0. (3!)n Prove that

A: We will prove it by Mathematical induction. First We show it is true for initial condition n=0…

Q: Prove that if n = nxn-1 holds integer value of -n, which means that for a positive integer value n,…

A: We have given that ddxxn=nxn-1 for any positive integer n. We need to show that, ddxx-n=-nx-n-1. We…

Q: show that no integer of the form n^3+1 is a prime other than 2

A:

Q: (24) Let p, q € N be two distinct primes. (a) Find the following quantities (possibly in terms of p…

A: Given: p, q∈ℕ are two distinct primes. To Do: (a) Find gcdp, q,   gcdp2q, pq2,   gcdpq, p2q2+1. (b)…

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

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.