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December 19, 2017Sieve of ElastensesIt is a very useful method for prime factorization. I will introduce it from the principle first. I will not mention the precise meaning of the words as it will be described in plain language. Natural numbers are made up of products of natural numbers. These natural numbers are called the factors of the first natural number. Let N be the initial natural number. Let's average N by product. In other words, let's try √.√ N.Here we get a factor of N greater than √ N. Letting it be a, the square of a is larger than N. Therefore, when dividing the factor of N into two, taking one factor larger than √ N, one becomes a factor smaller than √ N.For this reason, by dividing by the number √ N or less, N can be divided into two factors. If it can not be divided, N is a prime number. Repeat the same operation for the two factors you got in that way. Then repeat the operation until the factor becomes a prime number. By doing so, the number of factorizations decreases, so it is possible to operate quickly, which is very convenient.What we want to use here together is a method of judging whether 2, 3, 4, 5 and 9 are included in the factor. Mathematically it is terrible, but with easy words, one of the natural numbers has a factor of two, and one with three factors in the third. Therefore, this judgment is very useful.Is the last digit of even number 2 judgment method? If so, 2 is included as a factor. In the judgment method of 3, is it possible to divide the sum of each digit by 3? In the case of 4, can the lower two digits be divisible? 5 is that the last digit is 0 or 5.9 is that the sum of each digit is divisible by 9. 7 is a bit difficult. I will leave these commentaries to a number of sites.Let's try factoring 28938 into a trial.Focus on the last two digits. Oh, 4 will not break, but 2 will break.14469.Let's add each digit. 1 + 4 + 4 + 6 + 9 = 24. Alright. Break. When it is big that you add, add it and it is enough to think about it. 2 + 4 = 6. Yup. Break. By saying4823.This can not break even 2, 3 or 5. Activate an elastense sieve.√ 4823 = 69.44Let's think about prime numbers up to 70. I will try to divide by the next small 7. cracked!689.One more breath. √ 689 = 26.24 ... The next is 11 ... ... it will not break. So how about 13? did it.53.Finally √ 53 = 7.28 ... ... we divide 53 by the prime number up to 8 (up to 7) but it will not break. OK, 53 is a prime number.Factor factorization is complete.14469 = 2 × 3 × 7 × 13 × 53

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