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

Prove by induction that (2n+1)3 is an odd number for all positive integers n.

Prove that every prime greater than 3 can be written in the form 6n + 1 or 6n + 5.

there are no integers x and y such that x is a prime greater than 5 and x=6y+3

prove that it is not possible to find four distinct prime numbers p, q, r, s such that ps − qr = 0.

how would you prove this

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

5) Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4. In other words, prove that it is not 4 4 possible for n +m =4 k + 3, for integers n, m and k. [Hint: Consider fourth powers of even and odd integers.]

Use induction to prove that for every integer n  ≥ 0, 5n+2+ 62n+1 is divisible by 31.

Find all positive integers n that is a product of two distinct primes such that o(n) = 60.