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24-November-2007, 09:09 PM
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Prime numbers and double primes
I was reading through this thread (just started it, though) and it got me thinking about the randomness of primes. I figure there must be some form of pattern for them, even if we can't directly see it. The way I figure, if we start with all integers and subtract those that aren't primes, then we would be left with numbers that fall into the sets of
2n +/- 1 3n +/- 1 , 3n +/- 2 4n +/- 1 , 4n +/- 2 , 4n +/- 3 etc which basically just rids us of all forms of 2n, 3n, 4n, etc, so that all of the primes that are left should be equal to a number in each of the sets above, up to the value of the integer we are finding for. Well, anyway, I started working through that and realized that the set 6n +/- 1 is the only one that can be used for the set for 6, since 6n +/- 2 and 6n +/- 4 will each give a number divisible by 2, and 6n +/- 3 will give a number divisible by 3, and 6n +/- 5 is the same as 6n +/- 1 for a higher or lower value of n (both of which are also true for the set for 4 above). Since all primes must match a number in every set, then this means that all primes greater than 3 must be of the form 6n +/- 1. All double primes, then, which are separated from each other by two, must be of the form 6n - 1, 6n + 1. Is this known? I've looked around some but haven't found it. I'm sure it is but I'll ask anyway.  I've just started working on this but I'm wondering what else might be determined by such sets. Any suggestions?
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Let's put together the pieces of The Grand Puzzle . (website) "Let's define another operator, Sz, which we won't pay any attention to." "This transformation will automatically make zero equal zero." "It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time." 
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24-November-2007, 11:10 PM
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Primes have passed every test for randomness, which is why they've been at the core of encryption technology for many years.
However, because primes are reproducible on both sides, they've since been abandonded as keys, regardless of combination, as the assumed use of primes seriously reduces the keyspace of whatever encoding technology is used. Since the heyday of primes, we've developed unbreakable encryption. Doesn't matter the power of the computer, or how many billions of years you hack away at it. By it's very nature, it can't be done. And that's the way we do business today.
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I am Mugs, of the Alien clan of Usa, Nordamerica, a Terran, of Sol. "A positive attitude may not solve all your problems, but it will annoy enough people to make it worth the effort." - Herm Albright
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24-November-2007, 11:59 PM
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0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...
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25-November-2007, 12:32 AM
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Yes,
generally any prime number greater than any primorial is of the form (n*that-primorial) + or - 1. where n is some integer greater than 0. This has been known since antiquity. If I'm not mistaking it is also proven that the set of primes of the form n!-1 is infinite which is probably why the distributed computer programs which search for the biggest known prime number utilize`primes of the form. added: Just remembered it's more complicated than that: generally any prime number greater than any primorial is of the form (n*that-primorial) + q +/-1. where n is some integer greater than 0 and q is a primorial or 0 and q<(that-primorial). Or Something like that. I do remember something.
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"They reasoned that an object situated at the center and related equally to the extremes in every direction can have no impulse to move in any specific direction. In fact, they compared the situation of such an object with that of a man violently but equally hungry and thirsty, standing at the same distance from food and drink and unable to decide in which direction to move." - Aristotle Last edited by a1call; 25-November-2007 at 03:08 AM.
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25-November-2007, 01:16 AM
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Oh, okay. Thanks everybody. I just thought it was interesting. I'll still keep going through it for a while, though.
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Let's put together the pieces of The Grand Puzzle . (website) "Let's define another operator, Sz, which we won't pay any attention to." "This transformation will automatically make zero equal zero." "It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."
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25-November-2007, 01:30 AM
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basically the prfoblem boils down to the fact that int(a/b) can not be solved by basic mathematical operators. In other words int(2.45)=2 can not be represented using mathematical operators -,+, .... I am sure you understand.
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"They reasoned that an object situated at the center and related equally to the extremes in every direction can have no impulse to move in any specific direction. In fact, they compared the situation of such an object with that of a man violently but equally hungry and thirsty, standing at the same distance from food and drink and unable to decide in which direction to move." - Aristotle
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25-November-2007, 03:20 AM
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Well, I know what you mean about int(a/b), but as for the rest, maybe I just haven't gotten that far yet.
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Let's put together the pieces of The Grand Puzzle . (website) "Let's define another operator, Sz, which we won't pay any attention to." "This transformation will automatically make zero equal zero." "It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."
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27-November-2007, 05:08 AM
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Well, looks like I may be getting a little closer to an algorithm that can generate all primes and only primes, although I've still got quite a way to go. I have noticed that 6n +/- 1 will generate all primes and composites of two primes, but no composites containing three or more, and that is a very good start. But instead of 6n +/- 1, we can use 6n+1 and 6n+5, starting at n=0, which will give us all of the primes except for 2 and 3, the two multiples for the base number of 6n. The composites of two primes will contain a multiple of any two primes except for multiples of 2 and 3.
To limit this even further, then, instead of 6n, we can use an algorithm with a base number that contains more than two primes, such as 2*3*5=30. Then the list will be that of all primes and composites of any two primes except for those with multiples of 2, 3, and 5. We then add any number to 30n that does not have a common multiple with our base number, which for 30n, would be every prime up to 30 except for 2, 3, and 5. Below is an example for that algorithm, giving every prime (greater than 5) and composite of two primes (except for multiples of 2, 3, and 5) up to 1000. The solitary primes far out way the composites of two primes. There are 100 composites out of 266 in all, but the graph is still 62.4% pure primes, providing the first 166 prime numbers. Code:
30n n +1 +7 +11 +13 +17 +19 +23 +29 0 1 7 11 13 17 19 23 29 1 31 37 41 43 47 49 53 59 2 61 67 71 73 77 79 83 89 3 91 97 101 103 107 109 113 119 4 121 127 131 133 137 139 143 149 5 151 157 161 163 167 169 173 179 6 181 187 191 193 197 199 203 209 7 211 217 221 223 227 229 233 239 8 241 247 251 253 257 259 263 269 9 271 277 281 283 287 289 293 299 10 301 307 311 313 317 319 323 329 11 331 337 341 343 347 349 353 359 12 361 367 371 373 377 379 383 389 13 391 397 401 403 407 409 413 419 14 421 427 431 433 437 439 443 449 15 421 427 431 433 437 439 443 449 16 451 457 461 463 467 469 473 479 17 481 487 491 493 497 499 503 509 18 511 517 521 523 527 529 533 539 19 541 547 551 553 557 559 563 569 20 571 577 581 583 587 589 593 599 21 601 607 611 613 617 619 623 629 22 631 637 641 643 647 649 653 659 23 661 667 671 673 677 679 683 689 24 691 697 701 703 707 709 713 719 25 721 727 731 733 737 739 743 749 26 751 757 761 763 767 769 773 779 27 781 787 791 793 797 799 803 809 28 811 817 821 823 827 829 833 839 29 841 847 851 853 857 859 863 869 30 871 877 881 883 887 889 893 899 31 901 907 911 913 917 919 923 929 32 931 937 941 943 947 949 953 959 33 961 967 971 973 977 979 983 989 34 991 997
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Let's put together the pieces of The Grand Puzzle . (website) "Let's define another operator, Sz, which we won't pay any attention to." "This transformation will automatically make zero equal zero." "It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."
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27-November-2007, 05:43 AM
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Whether or not there will turn out to be some direct application for this particular endeavor, I'm not sure, but all of mathematics can be applicable in some aspect or other, I believe, and it caught my interest and I wasn't sure where else to put it. This part of the forum is labeled "general> questions and answers", but it looks like I'll probably just be working through it at this point the best I can. Any input would be helpful. If this needs to be moved, then that is fine too. Maybe 'general science' would have been better?
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Let's put together the pieces of The Grand Puzzle . (website) "Let's define another operator, Sz, which we won't pay any attention to." "This transformation will automatically make zero equal zero." "It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time." Last edited by grav; 27-November-2007 at 06:18 AM.
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27-November-2007, 04:09 PM
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27-November-2007, 04:35 PM
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The reason is that numbers of that special form has the fastest known primality test, which means a lot when you're looking at numbers where a normal pc can test one number in 3 months working all out. Currently largest is 232,582,657-1, with 9,808,358 digits.
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It would be hard to imagine an uglier building that hadn't won a major architectural award. Pratchett, Making Money Trying to make sense of computers, The Error Log.
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27-November-2007, 04:37 PM
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5 x 7 x 11 = 385 = 6 x 64 + 1 So your assertion that 6n +/- 1 excludes triple prime composites fails at the first place it could fail. Just because there is an algorithm, doesn't mean there is pattern. Chaos is generated by very simple formulas. In searching for a pattern in the primes, you are attempting what has occupied most of the finest mathematical brains for centuries. Indeed they have proven that there isn't a pattern in the normally accepted sense, as was said below. The Riemann Hypothesis is in effect the same subject. Read (at least) section 2 of this: http://en.wikipedia.org/wiki/Riemann_hypothesis You can win a lot of money by solving the Riemann Hypothesis. Last edited by Ivan Viehoff; 27-November-2007 at 04:40 PM. Reason: The thread was moved to a more appropriate category.
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27-November-2007, 06:04 PM
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Nice View From Up Here
At a little less difficult level, see if you can find a copy of George Gamow's book, "1,2,3 . . . Infinity", which among other things has a wonderful section on primes. And the advice of several previous posters is correct, put your effort where it will do the most good. Primes are like general and special relativity, but much much older; many many very bright people have explored the fundamentals already. Use the accumulated knowledge base. Stand on their shoulders and see what you can see.
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27-November-2007, 08:58 PM
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I saw something in Scientific American many
years ago that gave me to understand primes had been cracked! But it turned out just a curiousity, the Ulam spiral. Tried my own version from a corner and found an endless line of non-primes. Odd square followed by even number. Trivial presumably. There is plenty on Google about primes including many ways of picturing them. Here is a project, a histogram of the divisors from consecutive primes. I developed a special log base for the counting, nine columns between each power of ten. If you understand what I am describing!
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27-November-2007, 09:55 PM
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I've had a strong interest in primes since I was in high school. If you're looking for a pattern, then your best bet is to study the Riemann hypothesis. IIRC, there is a million dollar prize for the first person who can prove it true. I'm not sure if you get anything if you prove it false. Don't get your hopes up too fast though. As was mentioned before, some of the best mathematical minds have tried to prove it. These geniuses have made progress, but a definitive proof is still missing. Just understanding the Riemann hypothesis requires a good amount of college level complex analysis.
Over the years, I have accumulated a library of books about the subject, some good, some bad. Here is an abriged and annotated list (CYA disclaimer: I have no financial connection to any of these authors or publishers): Prime numbers and Computer Methods of Factorization IMO this is the best book on the subject that is accesible to someone with an undergraduate math background. Prime numbers; A computational Perspective Another good book that assumes an undergraduate math background written by one of the heavyweights in the field. Excursions in Number Theory This is the book that got me hooked on prime numbers 25 years ago. Written in the 1960's, it is very dated but accessible to anyone who remembers Algebra from high school. Factorization and Primality Testing. Well written book especially if you want to write any computer programs relating to prime numbers. This assumes a college level math aptitude. There are also several "layman's" books on the history of the Riemann Hypothesis. From a historical perspective, they may be interesting, but don't expect to learn any math from them.
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True is the question Derived from the Shakespeare Postulate (2b)|| !(2b)
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28-November-2007, 03:27 AM
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Let's put together the pieces of The Grand Puzzle . (website) "Let's define another operator, Sz, which we won't pay any attention to." "This transformation will automatically make zero equal zero." "It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."
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28-November-2007, 07:49 AM
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Ahh, prime numbers, I too have had a strong interest in them since I was in high school (and I just got out of high school). I think there is proof which says by definition prime numbers can't have a polynomial function which can predict all prime numbers. There are some curious ones like n^2 + n + 41, discovered by Euler which generates prime for all non-negative integers less than 40. There are many other curious functions like that on all levels and this is an area of interest to see how these functions are related and how to generate them. The ulam spiral is fascinating too. All prime numbers except 2 are odd numbers. Since in the Ulam spiral adjacent diagonals are alternatively odd and even numbers, it is no surprise that all prime numbers lie in alternate diagonals of the Ulam spiral. What is startling is the tendency of prime numbers to lie on some diagonals more than others. In effect I would have expected to see something like the "snow" on old TVs, but there are more patterns. About the 6n +/- 1 pattern It can be proven that in all possible values of a number mod 6 0 1 2 3 4 5 taking out numbers divisible by 2 or 3 (the prime numbers) you are left with, X 1 X X X 5 which are incidentally numbers of form 6n +/-1. Now this property is only because we are starting with the two smallest prime numbers and we "know" them. This shows all primes should be of this format. But not all numbers of this format are primes, obviously. We could even go on further with 2*3*5 = 30. |
