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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
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
1r(7,5) & r(5,7) [pc]12
2r(17,2) & r(2,17) [pc]21
3r(11,5) & r(5,11) [pc]21
4r(7,11) & r(11,7) [pc]25
5r(13,7) & r(7,13) [pc]27
6r(7,19) & r(19,7) [pc]33
7r(7,29) & r(29,7) [pc1] [pc2]43
8r(5,5) & r(19,19) [pc]43
9r(29,7) & r(7,29) [pc]43
10r(29,29) & r(7,7) [pc]65
11r(13,29) & r(29,13)84
12r(19,19) & r(31,31) [pc]100
13r(23,31) & r(31,23) [pc]108
14r(131,5) & r(5,131)146
15r(37,37) & r(43,43)160
16r(47,37) & r(37,47) [pc]168
17r(61,61) & r(43,43)208
18r(197,5) & r(5,197)212
19r(17,89) & r(89,17)212
20r(97,19) & r(19,97)232
21r(13,103) & r(103,13) [pc]245
22r(163,19) & r(19,163)383
23r(113,113) & r(37,37)413
24r(43,43) & r(131,131)479
25r(29,227) & r(227,29)541
26r(47,233) & r(233,47)607
27r(89,181) & r(181,89)629
28r(29,29) & r(197,197)649
29r(23,337) & r(337,23)743
30r(439,13) & r(13,439)917
31r(139,139) & r(211,211)1050
32r(89,89) & r(317,317)1129
33r(229,229) & r(149,149)1134
34r(53,499) & r(499,53)1157
35r(199,199) & r(191,191)1170
36r(379,379) & r(17,17)1171
37r(313,313) & r(107,107)1260
38r(149,337) & r(337,149)1458
39r(683,43) & r(43,683)1495
40r(439,439) & r(107,107)1638
41r(349,199) & r(199,349)1644
42r(571,571) & r(7,7)1720
43r(907,17) & r(17,907)1865
44r(907,29) & r(29,907)1901
45r(599,599) & r(59,59)1915
46r(631,631) & r(17,17)1927
47r(313,313) & r(347,347)1980
48r(139,139) & r(709,709)2544
49r(599,367) & r(367,599)2898
50r(617,617) & r(373,373)2970
51r(463,641) & r(641,463)3312
52r(971,971) & r(151,151)3366
53r(367,367) & r(811,811)3534
54r(631,631) & r(751,751)4146
55r(919,919) & r(587,587) [pc]4518

A000155 Prime Curios! Prime Puzzle
Wikipedia 155 Le nombre 155


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