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 by fractions p/q with p 0 and q 0 . In this variant of t
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). Clearly (p 1)! p 1 (mod p 1). Since
(p, p 1) = 1, we hav
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) (2)
(3) 27! 1 (mod 31), so 6 27! 1 (mod 31) and so 2
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) cfw_1, 1 and p = 2, it follows that (a/p) = (b/p).
(ii
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 (1 2i) = 5 1 (mod 4) and so 1 2i are Gaussian primes b
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, n (ie. m, n odd). 3 | m + n and 2 | m + n. 6 | m + n.
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 integer solutions of
2x + 5y = 99.
(1)
It is clear that x0
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 = 36 , so cannot be a Gaussian prime.
7. Note that N (1
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 (mod 9).
Reducing modulo the respective modulus, we ge
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 (6.7), then by (6.3)(i),
ord(g )
p1
4k
4k
=
=
=
= 4.
(k,
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 the rectangle. Solution : The continued fraction for 37/8
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 various sizes), and draw a gure showing how the 9 squa
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 labeled by fractions p/q with p 0 and q 0 . In this va
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 (a) Write down the product A1 A2 A3 A4 = 1 a1 1 a2 1 a
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 01 01 01 01 (a) Write down the product A1 A2 A3 A4 = 1 a
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 the Farey diagram across the edge 1/2, 1/3 (so in part
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 hence has determinant +1 , so we want to
use only matrice
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, then nding a linear fractional transformation T in LF (Z
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 triangles, then nding a linear fractional transformation
Math 3320 Problem Set 6
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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 7y 2 (b) 3x 2 4y 2 (c) x 2 + xy y 2
2. Using your answers in the
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 7y 2
(b) 3x 2 4y 2
(c) x 2 + xy y 2
2. Using your answ
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) Find the smallest positive nonsquare discriminant for w
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, it can obviously be written as dQ where Q is primitiv