Define a sequence by , and

for

For every and prove that divides

.

Suppose divides for some natural numbers and . Prove that divides

for

For every and prove that divides

.

Suppose divides for some natural numbers and . Prove that divides

Indian National Mathematical Olympiad 2010

Number Theory

Medium

Problem Solving Strategies by Arthur Engel

Do you really need a hint? Try it first!

For (a), use strong induction on .

The recurrence has the characteristic equation (see this post for details). This has two distinct roots, . Hence the general solution is of the form .

Solving for , we get . Suppose that , i.e. for some . Now it suffices to show that is an integer. Use induction on to prove this.

Note that the given expression is equivalent to . By induction, it suffices to show that is an integer. Write . Note that satisfies hence it suffices to prove that and are both integers. As , we are done.

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