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 certain 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 special FactorizationPreliminary effects the elemental Theorem of mathematics Euclid and the elemental Theorem of ArithmeticChapter Highlights difficulties functions of targeted Factorization A Puzzle Irrationality

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**Sample text**

Therefore, p | N − (N − 1) = 1, which is impossible. This means that p ≤ n is impossible, so we must have p > n. In particular, if n is prime, there is a prime p larger than n, so there is no largest prime. This means that there are infinitely many primes. CHECK YOUR UNDERSTANDING 3. Explain why 5 2 · 3 · 5 · 7 + 1. 3 Euclid’s Original Proof Here is Euclid’s proof that there is an infinite number of primes, using the standard translation of Sir Thomas Heath. Euclid’s statements are written in italics.

CHECK YOUR UNDERSTANDING 3. Explain why 5 2 · 3 · 5 · 7 + 1. 3 Euclid’s Original Proof Here is Euclid’s proof that there is an infinite number of primes, using the standard translation of Sir Thomas Heath. Euclid’s statements are written in italics. Since his terminology and notation may be unfamiliar, we have added comments in plaintext where appropriate. It will be helpful to know that when Euclid says “A measures B ” or “B is measured by A,” he means that A divides B or, equivalently, that B is a multiple of A.

13 implies that (a/d) divides (y0 − v). By definition, this means that there is an integer t with a y0 − v = t . 9), we get a b a (u − x0 ) = t . 11) d , we have a b u − x0 = t d or b u = x0 + t. 12), we have b u = x0 + t d and a v = y0 − t. 4), we have completed the proof. 4), we first verify that d | c. If it doesn’t, we’re done since there are no solutions. If it does, we divide both sides by d to get a new equation ax+by =c and in this equation, gcd(a , b ) = 1. For example, if we want to solve 6x+15y = 30, we divide by 3 and instead solve 2x+5y = 10.