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[ February 15, 2004 ]
ABBA primes
I would like to introduce here a new kind of digit-related primes.
For obvious reasons these are named ABBA primes and unlike many
others formats in my website they are not related to palindromes
(though the name might suggest they are...). Let me explain.
Take for example this curious example
777777777777713131313131313
Describe it and you will get something like
thirteen 7's followed by seven 13's.
Note that primes 7 and 13 occur twice. Once for the repeated
number string and secondly as the repeat index. Formally :
a b's followed by b a's
hence the abba name for this kind of primes
(and so nothing to do with Agnetha, Bjorn, Benny or Anni-frid).
Note that a and b must be prime, and a may equal b,
on the sides of the expression.
In fact we then have ( a a's followed by b b's )
I undertook a systematic search for all ABBA combinations
with primes between 2 and 1000. Here is the resulting
exhaustive list of all such primes and probable primes.
| Nr. | r(#a,"b") & r(#b,"a") | Length |
| 1 | r(7,5) & r(5,7) [pc] | 12 |
| 2 | r(17,2) & r(2,17) [pc] | 21 |
| 3 | r(11,5) & r(5,11) [pc] | 21 |
| 4 | r(7,11) & r(11,7) [pc] | 25 |
| 5 | r(13,7) & r(7,13) [pc] | 27 |
| 6 | r(7,19) & r(19,7) [pc] | 33 |
| 7 | r(7,29) & r(29,7) [pc1] [pc2] | 43 |
| 8 | r(5,5) & r(19,19) [pc] | 43 |
| 9 | r(29,7) & r(7,29) [pc] | 43 |
| 10 | r(29,29) & r(7,7) [pc] | 65 |
| 11 | r(13,29) & r(29,13) | 84 |
| 12 | r(19,19) & r(31,31) [pc] | 100 |
| 13 | r(23,31) & r(31,23) [pc] | 108 |
| 14 | r(131,5) & r(5,131) | 146 |
| 15 | r(37,37) & r(43,43) | 160 |
| 16 | r(47,37) & r(37,47) [pc] | 168 |
| 17 | r(61,61) & r(43,43) | 208 |
| 18 | r(197,5) & r(5,197) | 212 |
| 19 | r(17,89) & r(89,17) | 212 |
| 20 | r(97,19) & r(19,97) | 232 |
| 21 | r(13,103) & r(103,13) [pc] | 245 |
| 22 | r(163,19) & r(19,163) | 383 |
| 23 | r(113,113) & r(37,37) | 413 |
| 24 | r(43,43) & r(131,131) | 479 |
| 25 | r(29,227) & r(227,29) | 541 |
| 26 | r(47,233) & r(233,47) | 607 |
| 27 | r(89,181) & r(181,89) | 629 |
| 28 | r(29,29) & r(197,197) | 649 |
| 29 | r(23,337) & r(337,23) | 743 |
| 30 | r(439,13) & r(13,439) | 917 |
| 31 | r(139,139) & r(211,211) | 1050 |
| 32 | r(89,89) & r(317,317) | 1129 |
| 33 | r(229,229) & r(149,149) | 1134 |
| 34 | r(53,499) & r(499,53) | 1157 |
| 35 | r(199,199) & r(191,191) | 1170 |
| 36 | r(379,379) & r(17,17) | 1171 |
| 37 | r(313,313) & r(107,107) | 1260 |
| 38 | r(149,337) & r(337,149) | 1458 |
| 39 | r(683,43) & r(43,683) | 1495 |
| 40 | r(439,439) & r(107,107) | 1638 |
| 41 | r(349,199) & r(199,349) | 1644 |
| 42 | r(571,571) & r(7,7) | 1720 |
| 43 | r(907,17) & r(17,907) | 1865 |
| 44 | r(907,29) & r(29,907) | 1901 |
| 45 | r(599,599) & r(59,59) | 1915 |
| 46 | r(631,631) & r(17,17) | 1927 |
| 47 | r(313,313) & r(347,347) | 1980 |
| 48 | r(139,139) & r(709,709) | 2544 |
| 49 | r(599,367) & r(367,599) | 2898 |
| 50 | r(617,617) & r(373,373) | 2970 |
| 51 | r(463,641) & r(641,463) | 3312 |
| 52 | r(971,971) & r(151,151) | 3366 |
| 53 | r(367,367) & r(811,811) | 3534 |
| 54 | r(631,631) & r(751,751) | 4146 |
| 55 | r(919,919) & r(587,587) [pc] | 4518 |
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