Palindromic primes

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A palindromic prime

p

(in base

b

) is a palindromic number (in the given base) that is prime, and for which the palindrome (the number read backwards) is also prime. The first few palindromic primes in base 10 are (A002385)

{2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, ...}

In any base

b

, all primes

p < b

are trivially palindromic by virtue of having only one digit. If

b + 1

is [repunit] prime (as is the case with 11 in base 10), then it is the only palindromic prime with an even number of digits; all other palindromic numbers with an even number of digits (obviously repdigit numbers) are divisible by

b + 1

(a fact that follows naturally from the divisibility test for

b + 1

). In 2004, William Banks, Derrick Hart and Mayumi Sakata proved that almost all palindromic numbers are composite, and that in any base, as

x → ∞

,

#{n ∈ P (x) | n is prime} = O #P (x) log log log x log log x ,

where

P (x)

is the set of palindromic numbers up to

x

.

Table of palindromic primes (in base

b

)

b	A-number	Sequence
2	A016041	{3, 5, 7, 17, 31, 73, 107, 127, 257, 313, 443, 1193, 1453, 1571, 1619, 1787, 1831, 1879, 4889, 5113, 5189, 5557, 5869, 5981, 6211, 6827, 7607, 7759, 7919, 8191, 17377, 18097, ...}
3	A029971	{2, 13, 23, 151, 173, 233, 757, 937, 1093, 1249, 1429, 1487, 1667, 1733, 1823, 1913, 1979, 2069, 8389, 9103, 10111, 12301, 14951, 16673, 16871, 18593, 60103, 60913, 61507, ...}
4	A029972	{2, 3, 5, 17, 29, 59, 257, 373, 409, 461, 509, 787, 839, 887, 907, 991, 4289, 4561, 5189, 5669, 5861, 6133, 6217, 6553, 6761, 7309, 7517, 7789, 7853, 12899, 13171, 13591, 14327, ...}
5	A029973	{2, 3, 31, 41, 67, 83, 109, 701, 911, 1091, 1171, 1277, 1327, 1667, 1847, 2083, 2213, 2293, 2423, 2473, 2579, 2659, 2789, 2969, 3049, 16001, 16651, 19531, 20431, 21481, ...}
6	A029974	{2, 3, 5, 7, 37, 43, 61, 67, 191, 197, 1297, 1627, 1663, 1699, 1741, 1777, 1999, 2143, 2221, 2293, 2551, 6521, 6779, 7001, 7109, 7151, 7187, 7331, 7481, 7517, 7703, 47521, ...}
7	A029975	{2, 3, 5, 71, 107, 157, 257, 271, 307, 2549, 2647, 2801, 3347, 3697, 3851, 4201, 4397, 4649, 4951, 5399, 5651, 5749, 5903, 6449, 6701, 7451, 7703, 8053, 8501, 8753, 9103, ...}
8	A029976	{2, 3, 5, 7, 73, 89, 97, 113, 211, 227, 251, 349, 373, 463, 479, 487, 503, 4289, 4481, 4937, 5393, 5521, 5657, 5849, 6761, 7537, 7993, 12547, 12611, 12739, 13003, 13259, ...}
9	A029977	{2, 3, 5, 7, 109, 127, 173, 191, 227, 337, 373, 419, 601, 619, 683, 701, 719, 6967, 7129, 7867, 8443, 9181, 9343, 10333, 10657, 11071, 11971, 12547, 13033, 13367, 13691, 14843, ...}
10	A002385	{2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, ...}

See also

• Palindromic numbers
• Repunit primes