THEORY OF NUMBERS, Math 115 A Homework 1 Due Wednesday October 2

1. Use the well-ordering property to show that \sqrt{3} is irrational.

2. Find a formula for ∑_{k=1}^{n} k^{3}.

3. Find and prove a simple formula for the sum of the first n Fibonacci numbers with odd indices when n is a positive integers, i.e. find a formula for f_{1} + f_{3} + … + f_{2n−1}

4. Find the quotient and remainder in the division algorithm, with divisor 17 and dividend

a) 100, b) -44, c) 289

5. Show that if a is an integer, then 3 divides a^{3} − a.

6. Show that the n-th Fibonacci number f_{n} is divisible by 4 if and only if n is divisible by 6.