Problems & Puzzles: Puzzles

 Puzzle 679 Prime-proof odd integers “Prime-proof odd integers” are odd composite numbers not ending in 5 that remain composite even when a single decimal digit of the number is changed. For the origins of these numbers see: 1 & 2 Q. Can you find palindrome-Prime-proof odd integers other than : 707696707 and 998020899?

Contributions came from Giovanni Resta, Vicente Felipe, J. K. Andersen, Hakan Summakoglu & Emmanuel Vanteighem.

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Giovanni Resta wrote:

there are quite a lot of these numbers.
Up to 10^15 I counted 2078 of them.
In this range, the first are
707696707, 998020899, 13753335731, 31776967713, 32782028723
and the last are
998980717089899, 999183464381999, 999222626222999.

The first such numbers made only of zeros and ones are
101110100111001011101 and 1111011011001001001001101101111.

The smallest such number with 100 digits is
100000000000000000000000000000000000000000000219099091200000
0000000000000000000000000000000000000001.

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Vicente Felipe wrote:

13.753.335.731
32.782.028.723
33.215.451.233
39.571.617.593
71.351.715.317
90.459.895.409
91.318.881.319
93.374.247.339
96.207.570.269
96.482.728.469
96.898.089.869
99.407.270.499

Los más grandes que he podido encontrar son:
1.260.000.000.000.000.090.900.000.000.000.000.621
7.490.010.000.000.000.090.900.000.000.000.100.947

***

J. K. Andersen wrote:

The next examples: 13753335731, 31776967713, 33215451233, 39571617593.
A 2013-digit example: (10^2013-1)/9 + 65161816156*10^1001.

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Hakan Summakoglu wrote:

I tested integers <10^13 and found 189 palindrome-prime-proof odd integers.

There are 2 palindromes with 9 digits, 17 palindromes with 11 digits, 7 palindromes with 12 digits,

163 palindromes with 13 digits. Interestingly; 2,17,7,163 are prime numbers.

There are 2 palindromes with 9 digits:
707696707, 998020899

0 palindrome with 10 digits.

17 palindromes with 11 digits:
13753335731, 31776967713, 32782028723, 33215451233, 39571617593,
71351715317, 78220902287, 90459895409, 91318881319, 93374247339,
95692929659, 96207570269, 96482728469, 96898089869, 97559795579,
99407270499, 99532323599

7 palindromes with 12 digits:
112529925211, 300338833003, 759727727957, 778478874877, 905233332509,
912052250219, 976067760679

163 palindromes with 13 digits:
1038687868301, 1050926290501, 1083899983801, 1085909095801, 1129166619211,
1132789872311, 1134577754311, 1151616161511, 1201767671021, 1222482842221,
1254673764521, 1308308038031, 1390349430931, 1430907090341, 1450265620541,
1457431347541, 1519543459151, 1625118115261, 1645216125461, 1678900098761,
1687257527861, 1733757573371, 1760521250671, 1808055508081, 1829783879281,
1831845481381, 1835481845381, 1881387831881, 1899277729981, 1934104014391,
1964494944691, 1997645467991, 3044586854403, 3093928293903, 3177043407713,
3185737375813, 3226628266223, 3230005000323, 3244485844423, 3265063605623,
3356292926533, 3362890982633, 3383165613833, 3387678767833, 3405376735043,
3414337334143, 3424916194243, 3479835389743, 3490872780943, 3506731376053,
3527868687253, 3530558550353, 3539684869353, 3543957593453, 3563166613653,
3566618166653, 3580220220853, 3609018109063, 3633347433363, 3652568652563,
3673611163763, 3696068606963, 3699706079963, 3700469640073, 3764618164673,
3777495947773, 3792057502973, 3832603062383, 3853118113583, 3923335333293,
3934497944393, 3983803083893, 7040490940407, 7061642461607, 7124911194217,
7129054509217, 7159014109517, 7185087805817, 7185991995817, 7187320237817,
7203460643027, 7205678765027, 7243698963427, 7245877785427, 7320569650237,
7366452546637, 7367656567637, 7374136314737, 7412233322147, 7416108016147,
7433016103347, 7455998995547, 7460221220647, 7469279729647, 7487063607847,
7521964691257, 7527756577257, 7615940495167, 7679129219767, 7712734372177,
7725389835277, 7773915193777, 7780671760877, 7784257524877, 7821612161287,
7832486842387, 7851124211587, 7851195911587, 7882854582887, 7892653562987,
7902330332097, 7918111118197, 7940343430497, 7965577755697, 9002212122009,
9125718175219, 9134997994319, 9154202024519, 9170906090719, 9176489846719,
9180033300819, 9185644465819, 9202557552029, 9220240420229, 9246119116429,
9249524259429, 9308131318039, 9326425246239, 9330158510339, 9345728275439,
9369082809639, 9373958593739, 9392108012939, 9444308034449, 9450977790549,
9502456542059, 9504571754059, 9544802084459, 9577254527759, 9577643467759,
9593023203959, 9595167615959, 9596202026959, 9606735376069, 9616268626169,
9646574756469, 9653584853569, 9662882882669, 9710015100179, 9753066603579,
9755650565579, 9763635363679, 9781120211879, 9801589851089, 9831278721389,
9861583851689, 9865257525689, 9871968691789, 9886257526889, 9911614161199,
9944993994499, 9948707078499, 9978154518799.

***

Emmanuel Vanteighem

Here are all the palindromic prime-proof numbers below  10^14 :

707696707,998020899,13753335731,31776967713,32782028723,33215451233,39571617593,
71351715317,78220902287,90459895409,91318881319,93374247339,95692929659,96207570269,
96482728469,96898089869,97559795579,99407270499,99532323599,112529925211,300338833003,
759727727957,778478874877,905233332509,912052250219,976067760679,1038687868301,
1050926290501,1083899983801,1085909095801,1129166619211,1132789872311,1134577754311,
1151616161511,1201767671021,1222482842221,1254673764521,1308308038031,1390349430931,
1430907090341,1450265620541,1457431347541,1519543459151,1625118115261,1645216125461,
1678900098761,1687257527861,1733757573371,1760521250671,1808055508081,1829783879281,
1831845481381,1835481845381,1881387831881,1899277729981,1934104014391,1964494944691,
1997645467991,3044586854403,3093928293903,3177043407713,3185737375813,3226628266223,
3230005000323,3244485844423,3265063605623,3356292926533,3362890982633,3383165613833,
3387678767833,3405376735043,3414337334143,3424916194243,3479835389743,3490872780943,
3506731376053,3527868687253,3530558550353,3539684869353,3543957593453,3563166613653,
3566618166653,3580220220853,3609018109063,3633347433363,3652568652563,3673611163763,
3696068606963,3699706079963,3700469640073,3764618164673,3777495947773,3792057502973,
3832603062383,3853118113583,3923335333293,3934497944393,3983803083893,7040490940407,
7061642461607,7124911194217,7129054509217,7159014109517,7185087805817,7185991995817,
7187320237817,7203460643027,7205678765027,7243698963427,7245877785427,7320569650237,
7366452546637,7367656567637,7374136314737,7412233322147,7416108016147,7433016103347,
7455998995547,7460221220647,7469279729647,7487063607847,7521964691257,7527756577257,
7615940495167,7679129219767,7712734372177,7725389835277,7773915193777,7780671760877,
7784257524877,7821612161287,7832486842387,7851124211587,7851195911587,7882854582887,
7892653562987,7902330332097,7918111118197,7940343430497,7965577755697,9002212122009,
9125718175219,9134997994319,9154202024519,9170906090719,9176489846719,9180033300819,
9185644465819,9202557552029,9220240420229,9246119116429,9249524259429,9308131318039,
9326425246239,9330158510339,9345728275439,9369082809639,9373958593739,9392108012939,
9444308034449,9450977790549,9502456542059,9504571754059,9544802084459,9577254527759,
9577643467759,9593023203959,9595167615959,9596202026959,9606735376069,9616268626169,
9646574756469,9653584853569,9662882882669,9710015100179,9753066603579,9755650565579,
9763635363679,9781120211879,9801589851089,9831278721389,9861583851689,9865257525689,
9871968691789,9886257526889,9911614161199,9944993994499,9948707078499,9978154518799,
10358711785301,10590777709501,10858466485801,11388344388311,13957933975931,
14371311317341,14684455448641,14811177111841,15875444457851,18281299218281,
18323755732381,31002611620013,31113166131113,31885844858813,32855622655823,
33226988962233,33886455468833,34158311385143,34549511594543,35119466491153,
37757677675773,39893966939893,70038311383007,70429411492407,71419577591417,
71449888894417,71667711776617,72235622653227,73479111197437,73488500588437,
74606366360647,74607866870647,76290177109267,76342244224367,77698755789677,
77840299204877,78394077049387,79045511554097,79444011044497,79694877849697,
91048044084019,92653699635629,93592255229539,93600700700639,93727822872739,
93770800807739,94061955916049,94118844881149,94879166197849,95086155168059,
96129411492169,96206988960269,96270444407269,96390400409369,97651366315679,
97845777754879,99562611626599,99690811809699

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