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# Lecture_21 - MA1100 Lecture 21 Number Theory Fundamental...

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1 MA1100 Lecture 21 Number Theory Fundamental Theorem of Arithmetic Number of divisors Infinitude of primes Different types of p rimes Diophantine equations

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MA1100 Lecture 21 2 How many primes? Proposition There are infinitely many prime numbers. Proof By contradiction Suppose there are only finitely many primes. Let p 1 , p 2 , p 3 , ..., p m be all the primes. Consider the integer M = p 1 p 2 p 3 ...p m + 1 By FTA, M has a prime factorization Also M – 1 = p 1 p 2 p 3 ...p m
MA1100 Lecture 21 3 How many primes? • How many prime numbers are there that are congruent to 3 modulo 4 ? • How many prime numbers are there that are congruent to 3 modulo 15 ? If a and b are _____________ , then there are infinitely many prime numbers that are congruent to a modulo b . 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, …

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MA1100 Lecture 21 4 What is the largest prime?
MA1100 Lecture 21 5 Mersenne prime 2 n - 1 Marin Mersenne (1588-1648) If a Mersenne number 2 n -1 is a prime, we call it a Mersenne prime . Mersenne stated that, among the numbers 2 n -1 for n < 258, only the following are primes: n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 If 2 n - 1 is a prime, then n has to be a prime. Proposition

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MA1100 Lecture 21 6 Fermat prime Pierre de Fermat (1601-1665) What about primes of the form 2 n + 1 ? Fermat observed that, 2 2 n + 1 are primes for n = 0 ,1 ,2, 3, 4. If 2 n + 1 is a prime, then n has to be a power of 2. Proposition If a number of the form 2 n + 1 is a prime, we call it a Fermat prime .
MA1100 Lecture 21 7 Twin primes Definition Twin Prime Conjecture: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, ...

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Lecture_21 - MA1100 Lecture 21 Number Theory Fundamental...

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