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Number Theory Review 1

    Here is a list of 30 problems that will help a number theory student understanding their working knowledge and skill level for elementary number theory.

(1) Use mathematical induction to prove that whenever is a positive integer.

(2) Use mathematical induction to prove that for every positive integer

(3) Show that any amount of postage that is an integer number of cents greater than 53 cents can be formed using just 7-cent and 10-cent stamps.

(4) Use mathematical induction to prove that for

(5) Use mathematical induction to prove that for

(6) Find the Fibonacci number

(7) Find the Fibonacci number

(8) Show that whenever is a positive integer.

(9) Show that whenever is a positive integer.

(10) Show that whenever is a positive integer.

(11) Find the quotient and remainder in the division algorithm, (a) with divisor 17 and dividend (b) with divisor 17 and dividend

(12) What can you conclude if and are nonzero integers such that and

(13) Show that if is an integer, then divides

(14) Show that the product of every two integers of the form is of the form

(15) Use mathematical induction to show that the sum of the cubes of three consecutive positive integers is divisible by 9.

(16) Determine which of the following integers are prime:

    (a) 201   (b) 213   (c) 221   (d) 203   (e) 207   (f) 211

(17) Find all primes that are the difference of the fourth powers of two integers.

(18)  Show that no integer of the form is a prime, other than

(19) Find the smallest prime in the arithmetic progression where

    (a)    (b)    (c)
    

(20) Show that is prime for all integers with Show that it is composite for   

(21) Find the greatest common divisor of each of the following pairs of integers:

    (a)    (b) 0,100   (c)    (d)    (e) 100, 121   (f) 1001, 289

(22) Let b a positive integer. What is the greatest common divisor of and

(23) Show that if and are integers with then or 2.

(24) Show that if and are integers with then

(25) Show that and are relatively prime for all integers

(26) Use the Euclidean Algorithm to find the greatest common divisor and then write this as a linear combination of these integers.

(27) Use the Euclidean Algorithm to find the greatest common divisor and then write this as a linear combination of these integers.

(28) Use the Euclidean Algorithm to find the greatest common divisor and then write this as a linear combination of these integers.

(29) Find the prime factorization of

(30) Show that if and are positive integers and then

(31) Show that if and then

(32) Find the least common multiple of each of the following pairs of integers:

(33) Show that if and are integers, then if and only if and

Number Theory Books

A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of N... List Price: $18.95 Buy New: $7.50 You Save: $11.45 (60%) New (32) Used (23) from $7.50The Universe May Be a Mystery,But It's No SecretMichael Schneider leads us on a spectacular, lavishly illustrated journey along the numbers one through ten to explore the mathematical principles made visible (more)

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathem... List Price: $16.00 Buy New: $8.75 You Save: $7.25 (45%) New (38) Used (19) from $7.49In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled "On the Number of Prime Numbers Less Than a Given Quantity." (more)

Bayesian Computation with R (Use R) List Price: $49.95 Buy New: $37.86 You Save: $12.09 (24%) New (36) Used (13) from $35.00There has been a dramatic growth in the development and application of Bayesian inferential methods. Some of this growth is due to the availability of powerful simulation-based algorithms to summarize (more)

Euler's Gem: The Polyhedron Formula and the Birth of Topology List Price: $27.95 Buy New: $16.98 You Save: $10.97 (39%) New (31) Used (4) from $16.98Leonhard Euler's polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so (more)

e: The Story of a Number List Price: $19.95 Buy Used: $2.94 You Save: $17.01 (85%) New (11) Used (37) Collectible (1) from $2.94The interest earned on a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis are all intimately connected with the mysterious number e. In this informal (more)

Not Even Wrong: The Failure of String Theory and the Search for Unity in Phys... List Price: $16.95 Buy New: $7.61 You Save: $9.34 (55%) New (33) Used (11) from $7.27When does physics depart the realm of testable hypothesis and come to resemble theology? Peter Woit argues that string theory isn't just going in the wrong direction, it's not even science. Not Even Wrong (more)

Killer Poker By the Numbers: Mathematical Edge for Winning Play List Price: $14.95 Buy New: $6.99 You Save: $7.96 (53%) New (34) Used (13) from $6.99Killer Poker By the Numbers: Mathematical Edge for Winning Play (more)

Elementary Number Theory (5th Edition) List Price: $124.00 Buy New: $99.20 You Save: $24.80 (20%) New (10) Used (17) from $95.50Elementary Number Theory and Its Applications is noted for its outstanding exercise sets, including basic exercises, exercises designed to help students explore key concepts, and challenging exercises. (more)

Complex Analysis for Mathematics and Engineering List Price: $124.95 Buy New: $44.94 You Save: $80.01 (64%) New (14) Used (14) from $44.94Revised and updated, the new Fifth Edition of Complex Analysis for Mathematics and Engineering presents a comprehensive, student-friendly introduction to Complex Analysis. It's clear, concise writing (more)

Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills List Price: $29.95 Buy New: $14.75 You Save: $15.20 (51%) New (42) Used (16) from $14.49I used to think math was no fun'Cause I couldn't see how it was doneNow Euler's my heroFor I now see why zeroEquals e[pi] i+1--Paul Nahin, electrical engineer In the mid-eighteenth century, Swiss-born (more)