Math 3320 Problem Set 2
1
1. This problem involves another version of the Farey diagram, or at least the positive part of the diagram, the part consisting of the triangles whose vertices are labeled b
Solutions to Math 332 Homework 6
3. Given 1991! 1 (mod 1993). So (1993 1)! 1992 1991! 1 (mod 1993). By 3-22 (converse of
Wilsons theorem), 1993 is prime.
6. By Wilsons theorem, (p 1)! 1 p 1 (mod p). C
Solutions to Math 332 Homework 7
5. Since 899 = 29 31, we have that 28! 1 (mod 29) and 30! 1 (mod 31) by Wilsons theorem.
The former yields 1 27! 1 (mod 29) and so 27! 1 (mod 29). The latter yields (1
Solutions to Math 332 Homework 8
7. Note that (iii) actually follows immediately from denition.
(i) Given a b (mod p), we have
a
p
b
p
a(p1)/2 b(p1)/2
(mod p)
by Eulers criterion. Since (a/p), (b/p)
Solutions to Math 332 Homework 10
6. 390 = 2 3 5 13.
3 is a Gaussian prime by (11.11)(ii).
2 = (1 i)(1 + i); 1 + i = i(1 i) and so is a Gaussian prime by (11.11)(i).
5 = 12 + 22 = (1 2i)(1 + 2i); N
Solutions to Math 332 Homework 0 and 1
1. Using the results in the text, we have
(m, n) = 1 (m, n + m) = (n, n + m) = 1
by 1.22
(mn, n + m) = 1
by 1.10(i).
2. (m, 6) = (n, 6) = 3. So 3 | m, n and 2 m
Solutions to Math 332 Homework 2
30. 216k + 1 = (6k )3 + 1 = (6k + 1)(62k 6k + 1) is a composite number for every k 1 (note that the
factors are = 1).
36. We want to nd the number of non-negative inte
Solutions to Math 332 Homework 3
1. For to be a Gaussian prime, N ( ) must either be a rational prime (case (i) & (iii) of (11.11) or the
sqaure of a rational prime (case (ii) of (11.11). N ( ) = 729
Solutions to Math 332 Homework 5
29. 28x 6 (mod 70) has no solution since (28, 70) 6.
33. The system is
x 2
x5
x6
(mod 4)
(mod 7)
(mod 9).
So we need to solve
7 9b1 1
(mod 4),
4 9b2 1 (mod 7),
4 7b3 1
Solutions to Math 332 Homework 12
7. If a has order 4, then 4 = ord(a) | (p) = p 1. So p = 4k + 1 for some k N. Conversely if p = 4k + 1
for some k N, let g be a primitive root of p (this exists by (6
Math 3320 Prelim Solutions
Fall 2009
1. Determine how a 37 8 rectangle can be decomposed into 9 non-overlapping squares (of various sizes), and draw a gure showing how the 9 squares are arranged in th
Math 3320 Prelim
Fall 2009
No notes, books, calculators or other electronic devices are allowed during this exam. 1. Determine how a 37 8 rectangle can be decomposed into 9 non-overlapping squares (of
Math 3320 Problem Set 2 Solutions
1
1. This problem involves another version of the Farey diagram, or at least the positive part of the diagram, the part consisting of the triangles whose vertices are
Math 3320 Problem Set 3
1
1. This exercise is intended to illustrate the proof of the Theorem on page 15 of Chapter 1 in the concrete case of the continued fraction 1 + 1 + 1 + 1 . 2 3 4 5 01 01 01 01
Math 3320 Problem Set 3 Solutions
1
1. This exercise is intended to illustrate the proof of the Theorem on page 15 of Chapter 1 in the concrete case of the continued fraction 1 + 1 + 1 + 1 . 2 3 4 5 0
Math 3320 Problem Set 4
1
1. Find a formula for the linear fractional transformation that rotates the triangle 0/1, 1/2, 1/1 to 1/1, 0/1, 1/2 . 2. Find the linear fractional transformation that reects
Math 3320 Problem Set 4 Solutions
1
1. Find a formula for the linear fractional transformation that rotates the triangle 0/1, 1/2, 1/1 to 1/1, 0/1, 1/2 . Solution : A rotation preserves orientation he
Math 3320 Problem Set 5
1
For each of the rst six problems, compute the value of the given periodic or eventually periodic continued fraction by rst drawing the associated innite strip of triangles, t
Math 3320 Problem Set 5 Solutions
1
For each of the rst six problems, compute the value of the given periodic or eventually periodic continued fraction by rst drawing the associated innite strip of tr
Math 3320 Problem Set 6
1
1. Determine the periodic separator line in the topograph for each of the following quadratic forms (you do not need to include the fractional labels x/y ): (a) x 2 7y 2 (b)
Math 3320 Problem Set 6 Solutions
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1. Determine the periodic separator line in the topograph for each of the following quadratic forms (you do not need to include the fractional labels x/y ). (a) x 2
Math 3320 Problem Set 8
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1. Determine the number of equivalence classes of quadratic forms of discriminant = 120 and list one form from each equivalence class. 2. Do the same thing for = 61 . 3. (a)
Math 3320 Problem Set 9
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1. Preliminary comments: A quadratic form Q(x, y) = ax 2 + bxy + cy 2 is called primitive if the greatest common divisor of the coecients a, b, c is 1 . If Q is not primitive
Solutions to Math 332 Homework 11
1. We need only try the divisors of (37) = 36, starting from the smallest.
(a) 34 3 (mod 37), 342 9 (mod 37), 343 10 (mod 37), 344 7 (mod 37), 346 26
(mod 37), 349 1