MA 341: Introduction to Number Theory
Homework 3
Due June 5, 2015
10.1 - Let b1 < b2 < < b(m) be the integers between 1 and m that are relatively prime to m
(including 1), and let B = b1 b2 b3 b(m) be their product. The quantity B came up during the
proof

MTH 4436 Homework set 3.1, Page 44
Fall, 2004
Pat Rossi
Name
1. It has been mentioned that there are infinitely many primes of the form n2 2. Exhibit
five such primes.
n=
2
3
4
5
6
7
8
9
n2 2 = 2
2
7
14
23
34
47
62
71
Prime?
Yes!
Yes!
No!
Yes!
No!
Yes!
No

Problem 3.1
GROUP A
Final Draft (Jon K)
Rough Draft(Claudette F)
Working Notes (Anthony A and Emily D)
Extension/Presentation (Vincent L)
Page 1 of 4
A) If u and v have a common factor greater than 1, (a,b,c) will not be a primitive
Pythagorean triple.
We

Exercise 8.5
Working Notes: John Lucy, Tom Raymond
Rough Draft: Lauren Rizzotti
Final Draft: Abigail Bayer
Presentation: Ben Marlow
all incongruent solutions to each of the following linear congruences.
First we need to remember that in the forrnula ax =

Exercise Set 4
Math 2020
Due: March 20, 2007
Exercises to turn in:
1. In each case determine whether the statement is true or false. (A calculator will be
useful for the larger numbers.)
(a) 40 13 (mod 9)
(c) 8 48 (mod 14)
(e) 7754 357482 (mod 3643)
(b) 2

Math 223 Number Theory, Spring 07
Homework 5 Solutions
(1) Choose a number from 1 to 9 (inclusive). Multiply it by 3. Add 4 to the result. Multiply this by 3 and
subtract 8. Now add the digits of what you got. If the result is a two-digit number, add the

Math 223 Number Theory, Spring 07
Homework 4 Solutions
(1) Prove that all powers in the prime factorization of an integer n are even if and only if n is a perfect
square.
Solution: Let n have prime factorization
n = pa11 pa22 pa33 pann
If all ai are even,

Math 238: Solutions to Homework
Steven Miller (sjm1@williams.edu)
November 22, 2011
Abstract
Below are solutions / sketches of solutions to the homework problems from Math 238: Number
Theory (Smith College, Fall 2011, Professor Steven J. Miller, sjm1@will

ARITHMETIC PROGRESSIONS OF THREE SQUARES
KEITH CONRAD
1. Introduction
Here are the first 10 perfect squares (ignoring 0):
1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
In this list there is an arithmetic progression: 1, 25, 49 (common difference 24). If we search

Math 223 Number Theory, Spring 07
Homework 3 Solutions
(1) Use the definitions of an even and an odd integer to prove the following.
(a) The product of an even integer with any integer is an even integer.
(b) The sum of two odd integers is an even integer

HOMEWORK 01
MATH 4100
Problem 1.6:
For each of the following statements, fill in the blank with an
easy-to-check statement.
(a) M is a triangular number if and only if 8M + 1 is an odd
square.
(b) N is an odd square if and only if (N 1)/8 is triangular.
(

MATH3221: Number Theory
Homework until Test #2
Philipp BRAUN
Section 3.1
page 43, 1. It has been conjectured that there are infinitely many primes of the form n2 2.
Exhibit five such primes.
52
Solution. Five such primes are e.g. 2 = 4 2 = 22 2, 7 = 9 2 =

maﬂﬂﬂ
Page 1 ofq .
Group A
Final Draft: Claudette Foisy
Presentation: Jon Kaptcianos
Rough Draﬁ: Vinny Levesque
Working Notes: Anthony Aliquo
Emily Dowd
Problem 2.1 part A:
Show that either a or b of a Primitive Pythagorean Triple (a,b,c) must be a multip

Exercises for Chapter 5
1. a. Write a definition for gcd(a,b,c).
a. The greatest common divisor of the integers a, b, and c is that integer d such that d|a,
d|b, and d|c. If there exists any integer e such that e|a, e|b, and e|c, then e|d.
b. Use this def