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palprimes
A prime number  $p$  is a palprime if is it also palindromic, like 7, 101, or 1114111.

Clearly the property of being palindromic depends on the base. In base 10, all the palprimes except 11 have an odd number of digits, because all palindromes with an even number of digits are divisible by 11.

A few palindromic primes with palindromic index are knew. They are  $p_{1}=2$,  $p_{2}=3$,  $p_{3}=5$,  $p_{4}=7$,  $p{5}=11$,  $p_{8114118}=143787341$, and  $p_{535252535}=11853735811$.

The first palindromic primes (palprimes) are 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 more terms

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a-pointer 11 101 181 30203 + 938121839 955606559 967828769 aban 11 101 131 151 + 98000000089 98500000589 98600000689 alt.fact. 101 alternating 101 181 383 727 + 987898789 989252989 989898989 amenable 101 181 313 353 + 989898989 989919989 989969989 apocalyptic 787 929 10301 10501 + 19391 19891 19991 arithmetic 11 101 131 151 + 9980899 9981899 9989899 balanced p. 373 11411 30103 34543 + 990353099 994848499 996989699 c.decagonal 11 101 151 1598951 1128512158211 104216919612401 107635959536701 c.pentagonal 181 c.square 181 313 3187813 c.triangular 98689 Chen 11 101 131 181 + 9957599 9965699 9981899 congruent 101 151 181 191 + 9938399 9957599 9978799 Cunningham 101 Curzon 12821 14741 96269 1028201 + 187444781 187525781 187939781 cyclic 11 101 131 151 + 9980899 9981899 9989899 d-powerful 373 98389 3223223 3245423 + 9255529 9271729 9749479 de Polignac 373 757 11411 15551 + 9927299 9932399 9965699 deficient 11 101 131 151 + 9980899 9981899 9989899 dig.balanced 11 787 929 10301 + 198383891 198707891 199393991 economical 11 101 131 151 + 9980899 9981899 9989899 equidigital 11 101 131 151 + 9980899 9981899 9989899 esthetic 101 787 32323 34543 + 987676545676789 989878767878989 989898787898989 Eulerian 11 evil 101 353 373 383 + 999676999 999686999 999727999 good prime 11 101 191 727 + 166888661 176333671 195353591 happy 313 383 11311 15451 + 9916199 9926299 9935399 hex 919 Hogben 757 30103 Honaker 131 16661 33533 34543 + 971141179 981151189 982323289 iban 11 101 373 727 + 74747 77377 77477 inconsummate 383 16661 30703 37273 + 73037 73237 73637 Jacobsthal 11 junction 101 313 919 11311 + 9375739 9397939 9492949 Lucas 11 lucky 151 727 787 10501 + 9329239 9451549 9514159 m-pointer 131121131 1116111116111 magnanimous 11 101 nialpdrome 11 nude 11 oban 11 313 353 373 + 797 919 929 odious 11 131 151 181 + 999212999 999454999 999565999 Ormiston 1303031 1333331 1360631 1909091 + 977252779 977606779 977999779 palindromic 11 101 131 151 + 999999626999999 999999757999999 999999787999999 pancake 11 191 12721 106222601 panconsummate 11 353 pandigital 11 partition 11 101 pernicious 11 131 151 181 + 9935399 9965699 9981899 plaindrome 11 prime 11 101 131 151 + 99998189999 99998989999 99999199999 Proth 353 929 96769 repdigit 11 repunit 757 30103 self 727 929 30403 34543 + 997111799 997464799 998282899 sliding 11 101 star 181 12421 18481 121959676959121 314435969534413 396868131868693 strobogrammatic 11 101 181 18181 + 188881111188881 188881818188881 188888888888881 strong prime 11 101 191 727 + 9927299 9965699 9980899 super-d 131 181 919 10301 + 9822289 9896989 9907099 truncatable prime 313 353 373 383 + 79397 7693967 799636997 twin 11 101 151 181 + 998979899 999434999 999454999 uban 11 19000000091 32000000023 35000000053 + 7000009000007 9000005000009 9000007000009 Ulam 11 131 10501 13931 + 9752579 9787879 9871789 undulating 101 131 151 181 + 151515151515151 383838383838383 737373737373737 weak prime 131 151 181 313 + 9978799 9981899 9989899 weakly prime 79856965897 91507670519 Woodall 191 383 Zuckerman 11 zygodrome 11 1177711 7722277 7733377 + 999922222229999 999944222449999 999955444559999