[^^mac] [Back to MATHS page]Prime Twins
Prime twins (or twin primes if you prefer) are primes separated only by one number: for example: 11 and 13 are separated only by the number 12. Note that the first four primes (2, 3, 5, 7) form the ONLY possible prime qudruplette and as it turns out the only possible prime triplette. There are (apparently) an infinite number of prime twins. So, let's look at the possibility of prime tripplets. After 5, all multiples of 5 (10, 15, 20, 25, 30, 35) are multiples of 5 (and hence can never be prime). Toss out the evens, and you have the fact that ANY number > 5 that ends in 5 can't be prime; eg, 15, 25, 35, 45, 55, 65 ... hence the ONLY possible twin (or triplette) primes can end in "1, 3, 7, or 9". Cosider a sequence of odd numbers (excluding mutlitples of 5): x1 x3 x7 x9 y1 y3 y7 y9 (I show the sequence to consider sequences of "test" candidates), for example, consider: 21 23 27 29 31 33 37 39 (using the above sequence) -- -- -- -- I've marked the primes here, and there is in fact ONE twin prime (29, 31). Now, we think further: The ONLY possible sequence of prime triplettes (or quartettes) ends in" x7 x9 y1 y3 (since at each end of that sequence would again be a number ending in 5: 25 27 29 31 33 35 (to again use that section of -- -- prime "real estate") Now, we have to consider multiples of 3, which go like this: 3 6 9 12 15 18 21 24 27 30 33 36 ... Now let's look at a sequence like: x7 x9 y1 y3 (and try seeing where the "3's" multiples might lay)... x6 x7 x9 y1 y3 -- let it be just before the sequence starts, since x6 + 3 must be a multiple of 3, we end up x9 being a multiple of 3!! That is: x6 +3 -> x9 Erk! one of those ghastly composites!!!! Now, let x5 be the multiple of three: x5 x7 x9 y1 y3 -- x5 +3 -> x8 (still ok) x8 +3 -> y1 erk!! Now, let x4 be the multiple of three (our last chance) x4 x5 x7 x9 y1 y3 x4 +3 -> x7 is a multiple of 3 (but there's still the hope of 9 1 3) x7 +3 -> y0 is ok too y0 +3 -> y3 erk!!! Thus, there is no way to have any prime triplettes (except for the four muskateers at the head of the list ;) -- and all due to that nasty old three!!! The four muskateers: 2 3 5 7 "symetric twin-prime pairs" are marked with an "***" (ie, where they are separated ONLY by the number x5 or y0) "tropical islands" these consist of sequence of primes that form clusters of prime-twins (with no other primes in between them), indicated by xa xb-----ym yn-----zk zl 11 13 ***** 17 19 29 31 41 43 71 73 101 103 ***** 107 109 137 139 149 151 179 181 191 193 197 199 227 229 239 241 269 271 281 283 311 313 347 349 419 421 431 433 461 463 521 523 569 571 599 601 617 619 641 643 659 661 821 823*** **827 829 857 859 881 883 1019 1021 1031 1033 1049 1051 1061 1063 1091 1093 1151 1153 1229 1231 1277 1279 1289 1291 1301 1303 1319 1321 1427 1429 1451 1453 1481 1483 *** 1487 1489 1607 1609 1619 1621 1667 1669 1697 1699 1721 1723 1787 1789 1871 1873 *** 1877 1879 1931 1933 1949 1951 1997 1999 2027 2029 2081 2083 *** 2087 2089 2111 2113 2129 2131 2141 2143 2237 2239 2267 2269 2339 2341 2381 2383 2549 2551 2591 2593 2657 2659 2687 2689 2711 2713 2801 2803 2969 2971 2999 3001 3167 3169 3251 3253 *** 3257 3259 3299 3301 3329 3331 3359---3361------3371----3373------3389---3391 3461 3463*** *3467 3469 3527 3529 3539 3541 3557 3559 3581 3583 3671 3673 3767 3769 3821 3823 3851 3853 3917 3919 3929 3931 4001 4003 4019 4021 4091 4093 4127 4129 4157 4159 4217---4219------4229----4231------4241---4243 4259 4261 4271 4273 4337 4339 4421 4423 4481 4483 4517 4519 4547 4549 4637 4639 4649 4651 4721 4723 4787 4789 4799 4801 4931 4933--- -4967 4969 5009 5011------5021 5023 5059 5077 5099 5101 5107 5113 5119 5147 5153 5167 5171 5179 5189 5197 5209 5227 5231 5233 5237 5261 5273 5279 5281 5297 5303 5309 5323 5333 5347 5351 5381 5387 5393 5399 5407 5413 5417 5419 5431 5437 5441 5443 5449 5471 5477 5479 5483 5501 5503 5507 5519 5521 5527 5531 5557 5563 5569 5573 5581 5591 5623 5639 5641 5647 5651 5653 5657 5659 5669 5683 5689 5693 5701 5711 5717 5737 5741 5743 5749 5779 5783 5791 5801 5807 5813 5821 5827 5839 5843 5849 5851 5857 5861 5867 5869 5879 5881 5897 5903 5923 5927 5939 5953 5981 5987 6007 6011 6029 6037 6043 6047 6053 6067 6073 6079 6089 6091 6101 6113 6121 6131 6133 6143 6151 6163 6173 6197 6199 6203 6211 6217 6221 6229 6247 6257 6263 6269 6271 6277 6287 6299 6301 6311 6317 6323 6329 6337 6343 6353 6359 6361 6367 6373 6379 6389 6397 6421 6427 6449 6451 6469 6473 6481 6491 6521 6529 6547 6551 6553 6563 6569 6571 6577 6581 6599 6607 6619 6637 6653 6659 6661 6673 6679 6689 6691 6701 6703 6709 6719 6733 6737 6761 6763 6779 6781 6791 6793 6803 6823 6827 6829 6833 6841 6857 6863 6869 6871 6883 6899 6907 6911 6917 6947 6949 6959 6961 6967 6971 6977 6983 6991 6997 7001 7013 7019 7027 7039 7043 7057 7069 7079 7103 7109 7121 7127 7129 7151 7159 7177 7187 7193 7207 7211 7213 7219 7229 7237 7243 7247 7253 7283 7297 7307 7309 7321 7331 7333 7349 7351 7369 7393 7411 7417 7433 7451 7457 7459 7477 7481 7487 7489 7499 7507 7517 7523 7529 7537 7541 7547 7549 7559 7561 7573 7577 7583 7589 7591 7603 7607 7621 7639 7643 7649 7669 7673 7681 7687 7691 7699 7703 7717 7723 7727 7741 7753 7757 7759 7789 7793 7817 7823 7829 7841 7853 7867 7873 7877 7879 7883 7901 7907 7919 end. 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