Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.
Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.
Investigate Farey sequences of ratios of Fibonacci numbers.
Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?
If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?
What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?
Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?
By proving these particular identities, prove the existence of general cases.
With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?
This article gives an introduction to mathematical induction, a powerful method of mathematical proof.
