THEORY OF NUMBERS, Math 115 A Homework 1 Due Wednesday October 3 1. Use the well-ordering property to show that 3 is irrational. 2. Find a formula for
n k=1
k3 .
3. Find and prove a simple formula for the sum of the first n Fibonacci numbers with
THEORY OF NUMBERS, Math 115 A Homework 8 NOT Due Friday December 07 1. Find the sums and numbers of divisors of 196, 2100 and 10!. 2. Show that the number of divisors of n is odd if and only if n is a square. Characterize also the numbers whose sum o
THEORY OF NUMBERS, Math 115 A Homework 7 Due Friday November 30 1. Show that if c1 , . . . , c(m) is a reduced system of representatives modulo any m > 2 then c1 + + c(m) 0 (mod m). 2. Use Euler's theorem to easily conclude that 44444444 7 (mod
THEORY OF NUMBERS, Math 115 A Homework 6 Due Monday November 19 1. What is the remainder of 7 8 9 15 16 17 23 24 25 43 modulo 11? 2. Show that the periods in the decimal expansion of 1/83 and 1/1997 have at least 40 and 400 digits, respectiv
THEORY OF NUMBERS, Math 115 A Homework 5 Due Friday November 9 1. Find the solutions of each of the following systems of linear congruences: x 0 x0 x 1 x6 (mod (mod (mod (mod 2) 3) 5) 7) 3x 5 (mod 2) 5x 4 (mod 3) 2x 1 (mod 5) 5x 6 (mod
THEORY OF NUMBERS, Math 115 A Homework 4 Due Monday October 29 1. A student returning from Europe changes his French and Swiss francs into US. money. He only brought "whole francs", not fractions of them. He received 19 cents for each French and 59 c
THEORY OF NUMBERS, Math 115 A Homework 3 Due Friday October 19 1. Consider the "number system" {a + b -5 : a, b Z} (as a subset of the complex numbers. That is, -5 = 5i). It is usually denoted Z[ -5]. We define the "squared norm" N (a + b -5) =
THEORY OF NUMBERS, Math 115 A Homework 2 Due Wednesday October 12 1. Use the sieve of Eratosthenes to find all primes less than 250 2. Find all primes that are difference of the fourth powers of two integers. 3. Let Qn = p1 p2 .pn + 1 where p1 , . .