By Kraft, James S.; Washington, Lawrence C

ISBN-10: 1482214423

ISBN-13: 9781482214420

ISBN-10: 1931962022

ISBN-13: 9781931962025

IntroductionDiophantine EquationsModular ArithmeticPrimes and the Distribution of PrimesCryptographyDivisibilityDivisibilityEuclid's Theorem Euclid's unique facts The Sieve of Eratosthenes The department set of rules the best universal Divisor The Euclidean set of rules different BasesLinear Diophantine EquationsThe Postage Stamp challenge Fermat and Mersenne Numbers bankruptcy Highlights difficulties particular FactorizationPreliminaryRead more...

summary: IntroductionDiophantine EquationsModular ArithmeticPrimes and the Distribution of PrimesCryptographyDivisibilityDivisibilityEuclid's Theorem Euclid's unique facts The Sieve of Eratosthenes The department set of rules the best universal Divisor The Euclidean set of rules different BasesLinear Diophantine EquationsThe Postage Stamp challenge Fermat and Mersenne Numbers bankruptcy Highlights difficulties detailed FactorizationPreliminary effects the basic Theorem of mathematics Euclid and the basic Theorem of ArithmeticChapter Highlights difficulties purposes of targeted Factorization A Puzzle Irrationality

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**Extra resources for An Introduction to Number Theory with Cryptography**

**Sample text**

Since mathematicians like to prove the same result using different methods, we’ll give several other proofs of this result throughout the book. As you’ll see, each new proof will employ a different idea in number theory, reflecting the fact that Euclid’s theorem is connected with many of its branches. Here’s one example of an alternative proof. Another Proof of Euclid’s Theorem. We’ll show that for each n > 0, there is a prime number larger than n. Let N = n! + 1 and let p be a prime divisor of N .

There are also solutions such as x = −2, y = 5, which means you pay 5 quarters and get back 2 dimes. Before we get to the main result of this section, we look at two more examples that will help us understand the general situation. First, consider 6x − 9y = 20. Notice that 3 must divide the lefthand side but 3 is not a divisor of the right-hand side. This tells us that this equation can never have an integer solution. To make things notationally simpler, let d = gcd(a, b). We then see that in order for ax + by = c to have a solution, we must have d | c.

Then 0 = 5 · 0 + 0, so q = 0 and r = 0. CHECK YOUR UNDERSTANDING 5. Let a = 200, b = 7. Compute q and r such that a = bq + r and 0 ≤ r < b. 6. Let a = −200, b = 7. Compute q and r such that a = bq + r and 0 ≤ r < b. 1 The Division Algorithm 19 A Cryptographic Application Here’s an amusing cryptographic application of the Division Algorithm. Let’s say there is a 16-person committee that has to vote to approve a budget. The members prefer to keep their votes anonymous. Here’s a mathematical way to have every person vote Yes, vote No, or Abstain, while ensuring that all votes are kept secret.

### An Introduction to Number Theory with Cryptography by Kraft, James S.; Washington, Lawrence C

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