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Numbers  WON plate
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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  A000155 Prime Curios! Prime Puzzle  Wikipedia 155 Le nombre 155 ```

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Patrick De Geest - Belgium - Short Bio - Some Pictures