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36 37 neverendIng PrImes a proof by contradiction Just as every object of the real world can be split in a unique way into indivisible atoms, every natural number can also be written in a unique way as the product of indivisibles called primes the number 1 being an exception. The eight smallest primes are 2, 3, 5, 7, 11, 13, 17 and 19. The Sieve of Eratosthenes shown opposite is an elegant method for constructing all primes. Euclids Elements contains the following classic proof by contradiction that, unlike the real world, the world of numbers contains infinitely many primes. Proof There are either finitely or infinitely many primes. Assume that there are only finitely many and multiply all of them together to form a very large integer n 2 . 3 . 5 . 7... . Now, since n 1 is greater than any of the factors of n it cannot be prime, so one of the factors of n also has to be a factor of n 1. But, if this were so, then n 1 n 1 would also have the same factor. This is a contradiction, so we conclude that our assumption of finitely many primes must be false. Hence there are infinitely many prime numbers. Q.E.D. A twin of primes are two primes with a difference of two such as 5 7 and 11 13. Eternal fame awaits whoever can prove or disprove that there are infinitely many twins.
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