By Kraft, James S.; Washington, Lawrence C
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 designated FactorizationPreliminary effects the elemental Theorem of mathematics Euclid and the basic Theorem of ArithmeticChapter Highlights difficulties functions of detailed Factorization A Puzzle Irrationality. Read more...
summary: IntroductionDiophantine EquationsModular ArithmeticPrimes and the Distribution of PrimesCryptographyDivisibilityDivisibilityEuclid's Theorem Euclid's unique evidence 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 specified FactorizationPreliminary effects the elemental Theorem of mathematics Euclid and the elemental Theorem of ArithmeticChapter Highlights difficulties functions of distinctive Factorization A Puzzle Irrationality
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Extra resources for An Introduction to Number Theory with Cryptography
Show that 7 1005. 2 Euclid’s Theorem Fundamental to the study of the integers is the idea of a prime number. 5. A prime number is an integer p ≥ 2 whose only divisors are 1 and p. A composite number is an integer n ≥ 2 that is not prime. You may be wondering why 1 is not considered to be prime. After all, its only divisors are 1 and itself. Although there have been mathematicians in the past who have included 1 in the list of primes, nobody does so anymore. The reason for this is that mathematicians want to say there’s exactly one way to factor an integer into a product of primes.
Why is the process called a sieve? In our example, the multiples of the primes 2, 3, 5, 7 created a net. The numbers that fell through this net are the prime numbers. CHECK YOUR UNDERSTANDING 4. Use the Sieve of Eratosthenes to compute the prime numbers less than 20. 5 The Division Algorithm If a and b are integers, when we divide a by b we get an integer if and only if b | a. What can we say when b does not divide a? We can still make a statement using only integers by considering remainders. For example, we can say that 14 divided by 3 is 4 with a remainder of 2.
If a and b are integers with d = gcd(a, b), then gcd a b , d d = 1. Proof. If c = gcd(a/d, b/d), then c | (a/d) and c | (b/d). This means that there are integers k1 and k2 with a b = ck1 and = ck2 , d d which tells us that a = cdk1 and b = cdk2 . So, cd is a common divisor of a and b. Since d is the greatest common divisor and cd ≥ d, we must have c = 1. We’ll see later that calculating the greatest common divisor has important applications. So, it’s natural to ask, how do we go about finding the gcd when the answer is not immediately obvious?
An Introduction to Number Theory with Cryptography by Kraft, James S.; Washington, Lawrence C