Divisibility special

Divisibility special

Divisibility special

“Willaim Flash, king of the primes”. This very nice nick name was granted to me by Robert Fountain after my success with prime numbers in Gießen, November 2006. In this document I give explanation about my being so confident with prime numbers.

The history starts in 1954, when a mathematician tells me that a number is also divisible if it can be written as twice the sum of to squares on a different way. We’ll only discuss this theme now.

65 = 8² + 1² and 7² + 4². So it can be factorised and the prime factors themselves are the sum of 2 squares. Correct: 5 = 2² + 1² and 13 = 3² + 2².

From this time no – not so very big – number was safe for me. Owing to my profound knowledge of the squares up to 1.000 I could find out if a given number was twice the sum of 2 squares or not and if so by division find out the factors.

Eg 2993. A little bit tricky. Why? 93 can be composed as the sums of: 09+84, 29+64, 49+44, 69+24 and 89+04. Specially with numbers of 6 digits this is a lot of work. The way-out: to double and you get 5986. 86 can only be composed as the sum of 61 + 25, which means a considerable economisation of work. To reduce furthermore the quantity of work we take refuge to the 9-test. All the squares are (9) resp. 0, 1, 4, 7. On our search to the squares which can compose 5986=1(9) we have 19, 31, 69. We immediately can ignore 31, 31² being 7(9). And we find indeed: 5986 = 19² + 75² and 69² + 35². For finding the factors we reduce to resp. 47² + 28² by means of (47±28)/2 and 52² + 17² by means of (69±35)/2.

Finding the factors: take the even squares and do (52±28)/2 = 40 and 12. Take the odd squares and do (47±17)/2 = 32 and 15. 40 and 32 have common factor 8, 5× and 4×. 15 and 12 have common factor 3, 5× and 4×. First factor of the division 8³+3²=73. The other one 5²+4²=41.

This all brings us to the conclusion that – besides other criteria – a number certainly is prime if it can be written as only 1× the sum if 2 squares.

Is this all there is? No! Shortly later I thought about 3 combinations: 5×13×17, 1105. Well, here we find 4 combinations: the squares of resp. 4 + 33, 9 + 32, 23 + 24 and 32 + 9. This cried for a formula, which I found. The number of primes as described being n, the power is x, the number of combinations is n (x-1). The smallest possible combination is then 5×13×17×29= 32045. The amateurs can no find the 8 possible combinations of squares.

If 1 of the factors is square, eg 925, we have not 4 combinations, there are 3 combinations, of which surely 1 is with a 5. We find the square combinations 30 + 5, 22 + 21 and 27 + 14.

If 1 of the factors is ^3, 1625 = 5^3 × 13 we find 4 combinations, not 8, and 2 of them being with a 5. ( 40 + 5, 35 + 20, 37 + 16 and 29 + 28) .

To find the factors was always a lot of work. In a book of Dr. Rückle, Practice of calculating numbers, 1925. There I found important tips and now finding the factors is much easier.

I was given the phone number 20615346. I immediately saw 18 in it, so now we have 1145297. I started try and error and found 29 as a factor, division gave 39493, to double 78986= 2(9). We have to go the way of 19, 31, 69, 81 etc. The squares divisible by 3 can then be ignored, as 69, 81 etc. Also can be ignored the 19 etc squares with 161 and 561 as 25 squares with 425 and 825 do not exist .

The composing squares are: 281 + 5 and 181 + 215. The reduction: 143 + 138 and 198 + 17. Now we do (143±17)/2 and have 80 and 63. (198±138)/2 gives 168 and 30. We find as common factors for 168 and 63: 21, resp. 8 and 3 ×. For 80 and 30 we find 10, 8× and 3× too. So the factors of 39493 are 21²2 + 10² = 541 and 8² + 3² = 73.

In the meantime you will see that more than 50 years practising like this gives an enormous insight in the prime numbers.

Prime numbers are in daily practised used a.o. for the encryption of electronic messages. I do not know how difficult it is to find the factors of a number of 100 digits, which is the product of 2 primes of equal bigness. If we have a computer which can test 2.000.000.000.000 divisors in a second, it needs 1038 seconds to test. To get an idea of big numbers: a year is ± 3.107 seconds, men’s life is ± 2.1019 seconds. We can safely conclude the primes numbers asked for will never be found.

Willem Bouman

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