MAS 5215/001, MAT 4900/005 Number Theory, Spring 2013
Test 1
Date: 02/08/2013
Name:
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(1 pt for writing your name nicely)
50 points equal 100% for MAT 4900.
55 points equal 100% for MAS 5215
Question 1. (6 points) True or false: Explanation
MAS 5215/001, MAT4900/005 Number Theory, Spring 2013
CRN: 14108, 12734
Assignment 4 (Solution)
Question 1. (Pseudoprimes) (1 point) (a) Show that n = 645 is a not strong pseudoprime to base 2 by
showing that it does not pass Millers test (base 2).
(b) Sho
MAS 5215/001 Number Theory, Spring 2015
Test 1 (60 minutes)
Date: 02/05/2015
Name:
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50 points equal 100%.
(1 pt for writing your name nicely)
Question 1. (6 points) True or false: Explanation is not needed. Assume that m Z+ and
a, b, c, d,
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 5
Question 1.
(Exercise 3.7.) (1 point) For n Z+ , let (n) be the number of prime factors, counting
multiplicities, of n. For example (24 53 ) = 7 and (1) = 0. Dene the Liouville -function by
(n) = (1)(n) ,
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 1
Question 1. (Spectrum sequences and Beatty sequences) (4 points)
of is the sequence (n )n1 where
n = n.
Let R. The spectrum sequence
(a) Prove that if , R and = , then (n )n1 = (n )n1 .
1
1
(b) Assume for
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 4
Question 1. (The CRT and Wilsons Theorem.) (2 points) Suppose that k Z+ and both 4k + 1 and
4k + 3 are prime. Prove that
(4k)! 12k 2 + 11k + 1 (mod (4k + 1)(4k + 3).
Question 2. (Pseudoprimes.) (1 point) L
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 6 (Solution)
Question 1. (Dirichlet convolution. See also Exercises 3.68.) (1 point) Suppose that f and g are arithmetic
functions. Dene the Dirichlet product, or Dirichlet convolution as the function f g gi
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 2
Question 1. (1.9, generalized) (2 points) Let a Z.
(a) Prove that for nonnegative integers t, q, r,
gcd(atq+r 1, aq 1) = gcd(aq 1, ar 1).
(b) Prove that for nonnegative integers m, n,
gcd(an 1, am 1) = |ag
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 3
Question 1. (A divisibility test.) (2 points) Let n = 1000x + 100y + 10z + w.
(a) Find an inverse of 1000 modulo 31.
(b) Prove that 31 | n i 31 | (x 3y + 9z + 4w).
(c) Let n = 17, 612, 242, 776. Check if 3
MAS 5215/001 Number Theory, Spring 2015
Test 2 (60 minutes)
Date: 03/12/2013
Name:
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50 points equal 100%.
(1 pt for writing your name nicely)
Question 1. (3 points) True or false: Explanation is not needed. Assume that m, n Z+ and
b Z.
(a)
MAS 5215/001 Number Theory, Spring 2015
Test 3 (60 minutes)
Date: 04/23/2015
Name:
Show ALL steps.
(1 pt for writing your name nicely)
50 points equal 100%.
Question 1. (5+5 points) Assume that f, F are arithmetic functions.
(a) Assume that n 1. Prove tha
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 1 (Solution)
Question 1. (Spectrum sequences and Beatty sequences) (4 points)
of is the sequence (n )n1 where
n = n.
Let R. The spectrum sequence
(a) Prove that if , R and = , then (n )n1 = (n )n1 .
1
1
(b)
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 2 solution
Question 1. (1.9, generalized) (2 points) Let a Z.
(a) Prove that for nonnegative integers t, q, r,
gcd(atq+r 1, aq 1) = gcd(aq 1, ar 1).
(b) Prove that for nonnegative integers m, n,
gcd(an 1, am
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 6
Question 1. (Dirichlet convolution. See also Exercises 3.68.) (1 point) Suppose that f and g are arithmetic
functions. Dene the Dirichlet product, or Dirichlet convolution as the function f g given below:
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 3 (Solution)
Question 1. (A divisibility test.) (2 points) Let n = 1000x + 100y + 10z + w.
(a) Find an inverse of 1000 modulo 31.
(b) Prove that 31 | n i 31 | (x 3y + 9z + 4w).
(c) Let n = 17, 612, 242, 776.
MAS 5215/001, MAT4900/005 Number Theory, Spring 2013
CRN: 14108, 12734
Assignment 5 (Solution)
Question 1. (Generalized Fermat numbers) (1 point) It is easy to check (or see from class notes) that
if bm + 1 is prime and b > 1, then b is even and m = 2n fo
MAS 5215/001, MAT4900/005 Number Theory, Spring 2013
CRN: 14108, 12734
Assignment 6 solution
10 points equal 100%.
Question 1. (See also Exercise 4.5.) (1 point) Recall that we are concerned only with quadratic residues
relatively prime to the modulus m.
MAS 5215/001, MAT4900/001 Number Theory, Spring 2011
CRN: 114841,14842
Assignment 7 solution
Question 1. (Similar to Exercises 5.1, 5.2) (1 point)
Let m = 15. Find the orders of 1, 2, 4, 7, 8, 11, 13, 14 modulo m. Conclude that there is no
primitive root
MAS 5215/001, MAT4900/005 Number Theory, Spring 2013
CRN: 14108, 12734
Assignment 3 (Solution)
Question 1. (Divisibility Test) (1 point). Let n = 100x + 10y + z . Devise a divisibility test for divisor
67. Prove that your test is correct. Try out your tes
MAS 5215/001, MAT4900/005 Number Theory, Spring 2013
CRN: 14108, 12734
Assignment 2 (Solution)
(The extended Euclidean Algorithm) (1 point). Let a = 16663 and b = 14909. Find
Question 1.
gcd(a, b), s, t so that
gcd(a, b) = as + bt.
Solution. We start with
MAS 5215/001, MAT 4900/005 Number Theory, Spring 2013
Extra Credit for Exam 1 (10 points) (Solution)
Question 1. (Distributive properties.) (3 points) (a) Let e, e1 , e2 , . . . , e R and Z+ . Prove
one of the following identities:
maxcfw_mincfw_e, ei :
MAS 5215/001, MAT 4900/005 Number Theory, Spring 2013
Test 2
Date: 03/08/2013
Name:
Show ALL steps.
(1 pt for writing your name nicely)
50 points equal 100% for MAT 4900.
53 points equal 100% for MAS 5215
Question 1. (5 points) True or false: Explanation
MAS 5215/001, MAT 4900/005 Number Theory, Spring 2013
Test 3
Date: 04/24/2013
Name:
50 points equal 100% for all.
(1 pt for writing your name nicely)
Show ALL steps wuth explanation.
Question 1. (5+5 points) (a) Recall that (n), where n Z+ , is the number
MAS 5215/001, MAT4900/005 Number Theory, Spring 2013
CRN: 14108, 12734
Assignment 1 (Solution)
Question 1. (Spectrum sequences and Beatty sequences) (4 points)
of is the sequence (n )n1 where
n = n.
Let R. The spectrum sequence
(a) Prove that if , R and =
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 4 (Solution)
Question 1. (The CRT and Wilsons Theorem.) (2 points) Suppose that k Z+ and both 4k + 1 and
4k + 3 are prime. Prove that
(4k)! 12k 2 + 11k + 1 (mod (4k + 1)(4k + 3).
Solution. Let x (4k)! (mod (
MAS 5215/001, Spring 2015
CRN: 14560
Assignment 5 (Solution)
Question 1.
(Exercise 3.7.) (1 point) For n Z+ , let (n) be the number of prime factors, counting
multiplicities, of n. For example (24 53 ) = 7 and (1) = 0. Dene the Liouville -function by
(n)