17. Prove by induction that 1³+23++n³= (1+2+...+n)² for all positive integers n, that is, the sum of the cubes of the first n integers is equal to the square of the sum of the first n integers.

Show that the product of two numbers of the form 4n +1 is still of that form. Hence show that there are infinitely many primes of the form 4n + 3.

Prove or disprove: If three consecutive integers are multiplied together, and the second in order of size is added to the product, the result is always a perfect cube.

Prove that 2n>n2 if n is an integer greater than 4.

Prove: Any positive integer can be represented as an aggregate of different powers of 3, the terms in the aggregate being combined by the signs + and - appropriately chosen.

Prove that no number of form 4k+3 (where K is a positive integer) can ever be the sum of two perfect squares. Hint: think about adding two perfect squares.

Use the method of proof by cases to prove that for any integer n the product n(n + 1) is even.

(c) Prove that the product of k consecutive positive integers is divisible by k!

Prove that 2n + 3 ≤ 2n if n is an integer greater than 3.

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