Math 126 Number Theory
Prof. D. Joyce, Clark University
15 Mar 2006
Due Friday. Asmt. 10. From page 106: 2, 4, 5, 9;
The RSA algorithm is based on exponentiation
and from page 108 Misc. exercises: 8, 15.
modulo n. The encoding and decoding algorthims
are
Math 126 Number Theory
Prof. D. Joyce, Clark University
27 Feb 2006
Due Wednesday. Asmt. 8, , from page 76, exerSystems of linear congruences with the
cises 13, 79, 1113, 17, 20.
same modulus. The Chinese remainder theorem
deals with linear systems when t
Math 126, Number Theory
Second Test, alternate answers 11 Apr 2006
Problem 1. On Pythagorean triples. [18] Recall that a Pythagorean triple (x, y, z) consists of three positive integers such that x2 +y 2 = z 2 . Show that for any Pythagorean triple at lea
Math 126 Number Theory
Prof. D. Joyce, Clark University 3 Mar 2006
Due Today. Asmt. 9, from page 82, exercises 1, that is (aa1 )(aa2 ) . . . (aa(n) ). These two products 2, 4, 10; and from page 86, exercises 1, 2, 6, 7 have all the same terms since every
Math 126, Number Theory
Second Test Answers
April 2006
Scale. 85100 A, 7084 B, 5069 C. Median 70.
No. 16 and 10 are not relatively prime.
Problem 1. On Pythagorean triples. [18] Recall that a Problem 3. [18] Find at least two positive solutions of
Pythago
Math 126 Number Theory
Prof. D. Joyce, Clark University
20 Mar 2006
Due Friday. Page 148: 1, 4, and page 151: 3, 7, to have been the inspiration for Fermat to general11.
ize the problem. He likely began with the analogous
equation in three dimensions
Seco
Math 126 Number Theory
Prof. D. Joyce, Clark University
13 Mar 2006
Due Friday. Asmt. 10. From page 106: 2, 4, 5,
9; and from page 108 Misc. exercises: 8, 15. Treat
these last two as challenge problems. Well have
time on Wednesday for hints if you want th
Math 126 Number Theory
Prof. D. Joyce, Clark University
24 Apr 2006
Last time. Introduction to Pell equations. We (xn+1 , yn+1 ) is a solution of x2 dy 2 = cn+1 . One
saw some examples of how to nd solutions to Pell continued equation will do it.
equation
Math 126 Number Theory
Prof. D. Joyce, Clark University
28 Mar 2006
Due Friday. Page 155: exercises 1, 2, 7. Choose
one of the three and write it up completely.
Whichever one you choose, nd all those solutions
where the three numbers in the solution are r
Math 126 Number Theory
Prof. D. Joyce, Clark University
14 Apr 2006
A little Pythagorean music. Pythagoras studied musical pitch. If you take a single chord, a
monochord, and pluck it, one musical note, or pitch,
sounds. Well call that the fundamental. If
Math 126, Number Theory
Final Answers
May 2006
Scale. 85100 A. 70-84 B. 55 69 C. Median 86.
a. Choose two of the following numbers, call them m and
n, and show how the Euclidean algorithm is used to nd
their greatest common divisor d = (m, n).
Problem 1.