A Friendly Introduction to Number Theory

Joseph H. Silverman

Fourth Edition – ISBN: 978-0-321-81619-1 – © 2012 Pearson Education, Inc.
ix + 409 + (56 online) pages – Available from Amazon

A Friendly Introduction to Number Theory is an introductory undergraduate text designed to entice non-math majors into learning some mathematics, while at the same time teaching them how to think mathematically. The exposition is informal, with a wealth of numerical examples that are analyzed for patterns and used to make conjectures. Only then are theorems proved, with the emphasis on methods of proof rather than on specific results. Starting with nothing more than basic high school algebra, the reader is gradually led to the point of producing their own conjectures and proofs, as well as getting some glimpses at the frontiers of current mathematical research.

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Errata for the 4th edition. (Errata for the 3rd edition is also available.)

Friendly Introduction to Number Theory Book Jacket

Changes in the 4th Edition

There are a number of major changes in the fourth edition.

Some Numerical and Computer Exercises

To save time and eliminate typing errors, you can use this web page to copy and paste the numerical data for the following exercises.

Exercise 18.4 Here are two longer messages to decode if you like to use computers.
(a) You have been sent the following message:

5272281348, 21089283929, 3117723025, 26844144908, 22890519533,
26945939925, 27395704341, 2253724391, 1481682985, 2163791130,
13583590307, 5838404872, 12165330281, 28372578777, 7536755222.

It has been encoded using

p = 187963,     q = 163841,     m = pq = 30796045883,     and    k = 48611.

Decode the message.

(b) You intercept the following message, which you know has been encoded using the modulus

m = 956331992007843552652604425031376690367    and exponent    k = 12398737.

Break the code and decipher the message.

821566670681253393182493050080875560504,
87074173129046399720949786958511391052,
552100909946781566365272088688468880029,
491078995197839451033115784866534122828,
172219665767314444215921020847762293421.

Exercise 18.7 Write a computer program implementing one of the factorization methods that you studied in the previous exercise, such as Pollard's ρ method, Pollard's p-1 method, or the quadratic sieve. Use your program to factor the following numbers.
(a) 47386483629775753
(b) 1834729514979351371768185745442640443774091

Exercise 19.8 Program the Rabin-Miller test with multiprecision integers and use it to investigate which of the following numbers are composite.
(a) 155196355420821961
(b) 155196355420821889
(c) 285707540662569884530199015485750433489
(d) 285707540662569884530199015485751094149

Exercise 22.7 For this exercise, use the ElGamal cryptosystem described in Exercise 22.6.
(a) Bob wants to use Alice's public key a = 22695 for the prime p = 163841 and base g = 3 to send her the message m = 39828. He chooses to use the random number r = 129381. Compute the encrypted message (e1,e2) he should send to Alice.
(b) Suppose that Bob sends the same message to Alice, but he chooses a different value for r. Will the encrypted message be the same?
(c) Alice has chosen the secret key k = 278374 for the prime p = 380803 and the base g = 2. She receives a message (consisting of three message blocks)

(61745, 206881),    (255836, 314674),    (108147, 350768)

from Bob. Decrypt the message and convert it to letters using the number-to-letter conversion table in Chapter 18.

Interesting Number Theoretic Sites


Contact Information

No book is ever free from error or incapable of being improved. I would be delighted to receive comments, good or bad, and corrections from my readers. You can send mail to me at

jhs@math.brown.edu


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