I propose the new method of the prime numbers search in a natural scale. It differs from the existing one regarding the efficiency, as the prospector's tray from a dredge. The following is its essence. We classify the odd numbers of the natural scale to three sorts of numbers by three numerical filters casting aside all even numbers: (6п + 1), (6п + 3) and (6п + 5). We'll obtain three sorts of odd numbers.
The first sort of the natural scale odd numbers (6п + 1) 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103,109,115, 121 etc. The second sort of the natural scale odd numbers (6п + 3) 3, 9,15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 93, 99,105, 111,117, 123 etc. The third sort of the natural scale odd numbers (6п + 5) 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125 etc. There will be no prime numbers in the second sort of odd numbers except for the number 3 as the second numerical filter (6п + 3) = 3(2п + 1) won't let them pass there. All odd numbers of the second sort will be divided by the number 3. It diminishes the search area of prime numbers by a third in the "chaos" of the natural scale numbers.In addition, this method allows declining the Eratosthenes' method, at which it is necessary to let pass every explored number through the sieve practically "hand to hand". It allows creating the numerical series of the first and second sorts anew in the computer mode, tracking blank spaces in their line-up, in which only prime numbers can be disposed. It is possible to define the prime number value, which must be disposed in this gap as for the gap order number in the numerical series according to the following formulas:For the first sort of odd numbers: N1 = 6n1 - 5, where N1 is the numeral value of the first sort odd number, and n1 is its o! rder number in the first sort of odd numbers.For the second sort of odd numbers: N2 = 6n2 - 3, where N2 is the numeral value of the second sort odd number, and n2 is its order number in the second sort of odd numbers.For the third sort of odd numbers: N3 = 6n3 - 1, where N3 is the numeral value of the third sort odd number of the odd numbers, and n3 is its order number in the third sort of odd numbers.Its order number in every sort of odd numbers should be determined according to the following formulas as to the value of prime or else odd number.For the first sort of odd numbers: п1 = (N1 - 1): 6 + 1.For the second sort of odd numbers: п2 = (N2 - 3): 6 + 1.For the third sort of odd numbers: п3 = (N3 - 5): 6 + 1.The computer working as for this program can find prime numbers at any depth of the natural scale without human participation.THE WORK DESCRIPTION OF ODD NUMBERS THIRD BAND OF NATURAL SCALE To understand the given description, all mathematical dogmas should be forgotten for a time, in particular, Eratosthenes' sieve, as well as to consider the numerical series formation as the stretched process from time to time controlled by us.Let's imagine there is a narrow tape unfilled by numbers yet that lying before us. It has a beginning, but does not have an end. There is the numerical filter (п + 6), which divided it into the cells equal in size, at the beginning of this tape, having numbered each of them one after another.The numerical filter gave us out only five first prime numbers: 5, 11, 17, 23, 29. Let's try to fill in the whole numerical tape by these five numbers. We take the number 5 and put it in the first empty cell of the tape. Immediately after it the component numbers multiple to five appear instantly along the whole length of the endless tape being born by the number 5. Their multiplicity is provided by the fact that every new component number (except f! or the first) was disposed along the tape in the distance of five cells from its neighbor.After it the whole tape appeared being covered by the component numbers multiple to five "rooted to the ground" in it firmly, as if boundary posts. They will be the boundary reference points on the tape beginning to be filled by numbers. Let's pay attention to the fact that there will be four empty cells, unfilled by numbers yet, between all adjacent "posts", in which prime numbers can appear. It means that there can be no more than four adjacent prime numbers, non-parted by the component numbers, in this part of the numerical series.Setting the prime number 11 in the second empty cell we'll discover that the whole tape will appear filled by this number being born by new component numbers multiple to this number. Their multiplicity is provided by the fact that each of them is remote from each other on distance in eleven cells of the tape.Putting the prime number 17 into the third empty ! cell of the tape, the new component numbers multiple to 17, being born by this number, will appear on it instantly. Their multiplicity is guaranteed by the distance between them, which is equal to 17 of the tape cells. Having filled in the fourth empty cell of the tape by the prime number 23, the new component numbers multiple to the number 23 being born by this number will appear along its whole length at once. Their multiplicity is provided by the distance equal to the 23d tape cells between them as well.Setting the last prime number 29 being at our disposal into the fifth empty cell of the tape, the new component numbers being born by it will appear instantly along it, each of which will be multiple to the number 29. Their multiplicity is guaranteed by the distance between them, equal to 29 cells of the tape.Let's follow along the tape and observe what is occurring on it, getting free of the numbers given out by the numbers numerical filter to us. We'll discover that the ! fifth cell of the tape appeared filled-in by the component number 35, in spite of the fact that we did not put it there. But else the sixth cell of the tape appeared empty again. We calculated the number value being absent in it according to the formula N3 = 6n3 - 1, where n3 is the order number of the empty tape cell found out by us. It appeared equal to N3 = 41. We inserted this number into the seventh cell of the tape whereupon the new component numbers being born by it appeared multiple to the number 41 along the whole length of the tape, being in the distance of 41st cell from each other. At the further tape inspection we met both the cells filled-in by the component numbers, which we skipped, and empty cells, which we filled by the numbers obtained according to the formula given above. It is significant that the new component numbers multiple to this number being born by it appeared instantly along the whole tape after filling in every met empty cell by the calculated ! number. The distance between them always appeared equal to the amount of cells, determined by the value of the obtained number.All the numbers obtained by us, which we filled in, the empty cells found out, always were the PRIME numbers. And there were only the COMPONENT numbers in all cells already filled by the numbers, which we met in the process of our moving along the tape. And if so, it is possible to find the empty cells as well as ones filled by the numbers on the numerical tape at the computer moving of the third sort numbers of the natural series along the tape that is tantamount to the discovery of prime and component numbers without their verification as for the multipliers presence or absence in them. It is possible to discover the same in the first sort of the natural series numbers at the computer moving along the numerical tape. There are no prime numbers in the second sort of the natural series numbers except for the number 3.Therefore, it is possible to extract the prime number necessary for us from any depths of the natural series, available to the existent computers, by two computers moving along the first and third numerical tapes of the natural series numbers. We'll mark that everything can be carried out in the automatic mode without human participation. If you consider wholesome the use of the proposed method of the prime num! bers search in the natural series (observing my copyright), let me know to my e-mail: [EMAIL PROTECTED] It is desirable that the message would be in Russian.Best regards, K. Putro _______________________________________________ Prime mailing list [email protected] http://hogranch.com/mailman/listinfo/prime
