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 distinctive FactorizationPreliminary effects the basic Theorem of mathematics Euclid and the elemental Theorem of ArithmeticChapter Highlights difficulties functions of particular 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 precise FactorizationPreliminary effects the elemental Theorem of mathematics Euclid and the elemental Theorem of ArithmeticChapter Highlights difficulties functions of precise Factorization A Puzzle Irrationality
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Additional resources for An Introduction to Number Theory with Cryptography
This is a contradiction. Therefore G is not the same with any one of the numbers A, B, and C. And by hypothesis it is prime. Therefore the prime numbers A, B, C, and G have been found which are more than the assigned multitude of A, B, and C. Therefore, prime numbers are more than any assigned multitude of prime numbers. D. So G is a prime number that is not in our list of all possible primes, and so there can be no finite list of all primes. Therefore, there is an infinite number of primes. 4 The Sieve of Eratosthenes Eratosthenes was born in Cyrene (in modern-day Libya) and lived in Alexandria, Egypt, around 2300 years ago.
This means that our initial assumption that there is a finite number of primes must be incorrect. 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.
The fact that the sequence stops means that am must be prime, which means that am is a prime divisor of n. Example. In the proof of the lemma, suppose n = 72000 = 720 × 100. Take a1 = 720 = 10 × 72. Take a2 = 10 = 5 × 2. Finally, take a3 = 5, which is prime. Working backwards, we see that 5 | 72000. Euclid’s Theorem. There are infinitely many primes. Proof. We assume that there is a finite number of primes and arrive at a contradiction. 1) be the list of all the prime numbers. Form the integer N = 2 · 3 · 5 · 7 · 11 · · · pn + 1.