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[ *April 5, 2021* ]

Palindromic Primes from ninedigitals

by Alexandru Petrescu

We are looking for 19 digits palprimes obtained by concatenating

a ninedigital number with its reversal, but separated by a central

extra digit d.

Example: __145632987 2 789236541__

Obviously d can't be: 0, 3, 6 or 9.

May I present here the table with some statistics and the

lowest/greatest possible solutions.

Central digit | Number of Palprimes | The lowest ninedigital | The greatest ninedigital |
---|---|---|---|

14965 | 123458679 | 987653412 | |

13802 | 123459867 | 987654321 | |

14216 | 123456789 | 987653124 | |

15155 | 123456978 | 987654312 | |

14728 | 123456897 | 987654312 | |

14066 | 123456978 | 987654321 |

[ *March 8, 2005* ]

Palindromic primes embedded between two ones

by Zakir Seidov

Take 11 (the only 2-d palprime).

6. Now Q is shifted to 11-d palprimes.

Digits in PP | PP_pure | 1_PP_1 | 3_PP_3 | 7_PP_7 | 9_PP_9 |
---|---|---|---|---|---|

1 | 4 | 2 | ? | ? | ? |

3 | 15 | 3 | ? | ? | ? |

5 | 93 | 9 | ? | ? | ? |

7 | 668 | 93 | ? | ? | ? |

9 | 5172 | 421 | ? | ? | ? |

11 | 42042 | ? | ? | ? | ? |

Everyone with some spare time is invited to complete the above tabel.

Assignment _1_

Will the following ever happen again ?

Is the number **10501** a unique palindromic prime ?

When added together with its __preceding__ prime and its __following__ prime, the result is again a palindromic prime.

And as a pure coincidence we notice that :

that have a palindromic term in the sum somewhere.

[ *May 15, 2003* ]

Giovanni Resta (email) found eight more solutions to this assignment. Well done!

It pleases me a lot that this old puzzle (five years since its first publication!) still grabs

someone's attention. Giovanni scanned exhaustively all possible solutions

between 10501 and 324456535654423. The list now goes as follows :

[ *June 10, 2003* ]

Jens Kruse Andersen (email) found many big solutions and could find thousands

of "small" solutions (with 19-39 digits) in a day.

For all the details of his clever approach I refer to
Carlos Rivera's webpage at http://www.primepuzzles.net/puzzles/puzz_007.htm.

Here is just one solution (303 digits) from his email that

I prefer above the others because of its palindromic length !

Jens's current record is a solution with 527 digits. Amazing!p= 10^{302}+ 10*(7*10^{301}-70)/99 + 10^{140}*22001012200100221010022 + 1

Let w be "0" followed by 69 concatenations of "70".

Thenpis the concatenation 1w92708082907170928080729w1Solution equation: ( p-138) + (p) + (p+158) = (3p+20)These 4 primes were proved with Marcel Martin's Primo.

Assignment _2_

Can you find larger palindromic primes than

9918199 |
3306049 + 3306059 + 3306091 |

that is the sum of three or __more__ consecutive primes ?

They were all sent to me by email starting from [ *August 26 & September 28, 1998* ].

Note that some palindromic primes have very interesting patterns !

**11155555111** - a generalized __plateau__ prime - is expressible as 3 and 7 consecutive primes (see table (!)).

See also Carlos Rivera's ** Puzzle nr. 23** : Pal-primes adding consecutive primes.

[ *June 10, 2003 *]

Jens Kruse Andersen (email) found three titanic solutions (1001 digits).

For the strategy of his clever approach I refer to
Carlos Rivera's webpage at http://www.primepuzzles.net/puzzles/puzz_023.htm.

The solutions to assignment 1 are also valid here, but a search designed for

this problem can find bigger solutions.

Here is the second solution of his triplet :

Impressive, to say the least, Jens !p= 10^{1000}+ 1308107018031*10^{494}+ 1

Solution equation: ( x–3772) + (x–2094) + (x+5868), wherex= (p–2)/3The palprimes were proved with PrimeForm/GW and all the other primes with Marcel Martin's Primo.

Assignment _3_

Try extending one or more of the following seven lists

Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :reflectable primes. under A007616.Reflectable emirps. under A007628.reversal is prime. under A007500.reversal is a different prime). under A006567.P. De Geest] entries to the table.Click here to view some entries to the table about palindromes. |

The site The largest known primes keeps monthly updates also about palindromic primes.

Note that the length of this giant is palindromic too (no coincidence!).

The largest one in the list [ dating from *April 28, 1999* ] is **10^30802 + 1110111*10^15398 + 1** and is __30803__ digits long !

Harvey Dubner published various articles
about palindromic primes in the Journal of Recreational Mathematics.

Most of Harvey's record palindromic primes are on display.

The Top Ten by Rudolph Ondrejka

In Keith Devlin book All the Math that's Fit to Print chapter 92 we read about the largest known palindromic prime number (back in 1987 !).

It starts and ends with a 1, has a single 5 in the middle, and zeros everywhere else for a total of 2.977 digits.

Here is an excerpt from Martin Gardner's book

*"Puzzles from other Worlds"* page 107 :

*"The smallest palindrome prime containing all the ten digits is 1023456987896543201,
which was proved by Harry L. Nelson in 1980.
The largest known palindromic prime, discovered by Hugh C. Williams in 1977,
consists of the digit 1 repeated 317 times. It is called a repunit prime. The only
other known repunit primes are 11, and the primes formed by 19 and 23 units.
The number formed with 1,031 units is probably the next larger repunit prime,
but this has not yet been proved."*

Source of picture Pour La Science - avril 1999 - Logique et calcul - Les chasseurs de nombres premiers - (Jean-Paul Delahaye)

I'd like to state here that since then R(1031) is found to be 100% prime.

I quote from Eric Weisstein's Repunit page

*"The only base-10 repunit Primes R(n) for are n=2, 19, 23, 317, and 1031
(Sloane's A004023; Madachy 1979, Williams and Dubner 1986, Ball and Coxeter 1987, Granlund).
T. Granlund completed a search up to 45,000 in 1998 using two months of CPU time on a parallel computer."*

- A002385 Palindromic primes. - njas,sp
- A006341 Octal palindromes which are also primes. - James Carlson
- A007500 "Palindromic" primes: reversal is prime (the name is not accurate but I am using it because Wells did). - njas,rgw
- A007616 Palindromic reflectable primes. - njas,rgw,mb
- A016041 Palindromic primes in base 2. - Robert G. Wilson v
- A016115 Number of prime palindromes with n digits. - Robert G. Wilson v
- A028989 Smallest palindromic prime with 2n+1 digits. - Patrick De Geest
- A028990 Largest palindromic prime with 2n+1 digits. - Patrick De Geest
- A029732 Palindromic primes in base 16. - Patrick De Geest
- A029971 Palindromic primes in base 3. - Patrick De Geest
- A029972 Palindromic primes in base 4. - Patrick De Geest
- A029973 Palindromic primes in base 5. - Patrick De Geest
- A029974 Palindromic primes in base 6. - Patrick De Geest
- A029975 Palindromic primes in base 7. - Patrick De Geest
- A029976 Palindromic primes in base 8. - Patrick De Geest
- A029977 Palindromic primes in base 9. - Patrick De Geest
- A029978 Palindromic primes in base 11. - Patrick De Geest
- A029979 Palindromic primes in base 12. - Patrick De Geest
- A029980 Palindromic primes in base 13. - Patrick De Geest
- A029981 Palindromic primes in base 14. - Patrick De Geest
- A029982 Palindromic primes in base 15. - Patrick De Geest
- A030150 Palindromic primes in which parity of digits alternates. - Patrick De Geest
- A030286 Palindromic primes whose digits do not appear in previous term. - Patrick De Geest
- A030462 Palindromic prime concatenated with next palindromic prime is a prime. - Patrick De Geest
- A030463 Previous palindromic prime concatenated with this palindromic prime is prime. - Patrick De Geest
- A030464 Prime is a concatenation of two consecutive palindromic primes. - Patrick De Geest
- A030521 Product of first n palindromic primes plus 1. - Felice Russo
- A030522 Product of first n palindromic primes minus 1. - Felice Russo
- A030680 Smallest nontrivial extension of nth palindromic prime which is a prime. - Patrick De Geest
- A030681 Smallest nontrivial extension of nth palindromic prime which is a square. - Patrick De Geest
- A030683 Smallest nontrivial extension of nth palindromic prime which is a cube. - Patrick De Geest
- A030463 Previous palindromic prime concatenated with this palindromic prime is prime. - Patrick De Geest
- A031881 Lucky numbers both palindromic and prime. - Patrick De Geest
- A032350 Palindromic composite numbers (sort of complement of the palindromic primes). - Patrick De Geest
- A032593 Twin palindromic primes (lower terms). - Patrick De Geest
- A032594 Twin palindromic primes (upper terms). - Patrick De Geest
- A032595 Three consecutive palindromic primes (lower terms). - Patrick De Geest
- A032596 Three consecutive palindromic primes (middle terms). - Patrick De Geest
- A032597 Three consecutive palindromic primes (upper terms). - Patrick De Geest
- A033620 All prime factors are palindromes. - njas
- A033938 Palindromic primes n such that the period of 1/n is a palindrome. - Jud McCranie, G. L. Honaker, Jr.
- A035067 n! has a palindromic prime number of digits. - Patrick De Geest
- A035068 Palindromic prime lengths of factorials: see A035067. - Patrick De Geest
- A037010 Differences between adjacent palindromic primes. - G. L. Honaker, Jr.
- A039657 Number of digits in all (2n+1)-digit palindromic primes. - Enoch Haga
- A039679 Palindromic and prime Fibo-lucky numbers. - Felice Russo
- A039944 Undulating palindromic primes of form [ AB ]nA with alternating prime and non-prime digits. - G. L. Honaker, Jr.
- A039954 Palindromic primes formed from the reflected decimal expansion of pi. - G. L. Honaker, Jr.
- A040025 Number of prime palindromes with 2n+1 digits. - Patrick De Geest
- A045336 Palindromic prime primes: palindromic terms from A019546.- Robert G. Wilson v
- A045978 Palindromic primes that are "near miss circular primes" [all cyclic shifts except 1 are primes].- Carlos B. Rivera
- A046210 Smallest palindromic prime that generates a palindromic prime pyramid of height n. - Felice Russo
- A046349 Composite numbers with only palindromic prime factors. - Patrick De Geest
- A046350 Odd composite numbers with only palindromic prime factors. - Patrick De Geest
- A046351 Palindromic composite numbers with only palindromic prime factors. - Patrick De Geest
- A046355 Numbers with only palindromic prime factors whose sum is palindromic (counted with multiplicity). - Patrick De Geest
- A046356 Odd numbers with only palindromic prime factors whose sum is palindromic (counted with multiplicity). - Patrick De Geest
- A046357 Palindromes with only palindromic prime factors whose sum is palindromic (counted with multiplicity). - Patrick De Geest
- A046366 Numbers divisible by the palindromic sum of its palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046367 Odd numbers divisible by the palindromic sum of its palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046368 Numbers with exactly 2 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046369 Numbers with exactly 3 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046370 Numbers with exactly 4 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046371 Numbers with exactly 5 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046372 Odd numbers with exactly 2 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046373 Odd numbers with exactly 3 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046374 Odd numbers with exactly 4 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046375 Odd numbers with exactly 5 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046376 Palindromes with exactly 2 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046377 Palindromes with exactly 3 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046378 Palindromes with exactly 4 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046379 Palindromes with exactly 5 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046380 Palindromes with exactly 6 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046381 Palindromes with exactly 7 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046382 Palindromes with exactly 8 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046383 Palindromes with exactly 9 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046384 Palindromes with exactly 10 palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046385 Smallest palindrome with exactly n palindromic prime factors (counted with multiplicity). - Patrick De Geest
- A046400 Numbers with exactly 2 distinct palindromic prime factors. - Patrick De Geest
- A046401 Numbers with exactly 3 distinct palindromic prime factors. - Patrick De Geest
- A046402 Numbers with exactly 4 distinct palindromic prime factors. - Patrick De Geest
- A046403 Numbers with exactly 5 distinct palindromic prime factors. - Patrick De Geest
- A046404 Odd numbers with exactly 2 distinct palindromic prime factors. - Patrick De Geest
- A046405 Odd numbers with exactly 3 distinct palindromic prime factors. - Patrick De Geest
- A046406 Odd numbers with exactly 4 distinct palindromic prime factors. - Patrick De Geest
- A046407 Odd numbers with exactly 5 distinct palindromic prime factors. - Patrick De Geest
- A046408 Palindromes with exactly 2 distinct palindromic prime factors. - Patrick De Geest
- A046409 Palindromes with exactly 3 distinct palindromic prime factors. - Patrick De Geest
- A046410 Palindromes with exactly 4 distinct palindromic prime factors. - Patrick De Geest
- A046472 Palindromic primes in base 10 and base 2. - Patrick De Geest
- A046473 Palindromic primes in base 10 and base 3. - Patrick De Geest
- A046474 Palindromic primes in base 10 and base 4. - Patrick De Geest
- A046475 Palindromic primes in base 10 and base 6. - Patrick De Geest
- A046476 Palindromic primes in base 10 and base 7. - Patrick De Geest
- A046477 Palindromic primes in base 10 and base 8. - Patrick De Geest
- A046478 Palindromic primes in base 10 and base 9. - Patrick De Geest
- A046479 Palindromic primes in base 10 and base 11. - Patrick De Geest
- A046480 Palindromic primes in base 10 and base 12. - Patrick De Geest
- A046481 Palindromic primes in base 10 and base 13. - Patrick De Geest
- A046482 Palindromic primes in base 10 and base 14. - Patrick De Geest
- A046483 Palindromic primes in base 10 and base 15. - Patrick De Geest
- A046484 Palindromic primes in base 10 and base 16. - Patrick De Geest
- A046485 Sum of first n palindromic primes. - Patrick De Geest
- A046490 Palindromes expressible as the sum of 2 consecutive palindromic primes. - Patrick De Geest
- A046491 Palindromes expressible as the sum of 3 consecutive palindromic primes. - Patrick De Geest
- A046492 Palindromic primes expressible as the sum of 3 consecutive palindromic primes. - Patrick De Geest
- A046493 Primes expressible as the sum of 3 consecutive palindromic primes. - Patrick De Geest
- A046705 Palindromic primes whose product of digits is a prime. - Felice Russo
- A046852 Numerator of sum of reciprocals of first n palindromic primes. - G. L. Honaker, Jr.
- A046853 Denominator of sum of reciprocals of first n palindromic primes. - G. L. Honaker, Jr.
- A046941 Primes p(n) which are palindromes and their indices n are also palindromes. - Carlos B. Rivera F.
- A047076 a(n+1) is the smallest palindromic prime containing exactly 2 more digits on each end than the previous term, with a(n) as a central substring. - G. L. Honaker, Jr.
- A048052 Start of the first occurrence of n consecutive reversible primes (emirps). Palindromic primes are allowed. - Jud McCranie
- A048662 Dihedral palindromic primes. - Felice Russo
- A048677 Concatenation of first n palindromic primes. - Felice Russo
- A048796 Palindromic primes formed from decimal expansion of pi written backwards then forwards. - G. L. Honaker, Jr.
- A050236 Indices of consecutive squares palindromic primes; x such that x^2+(x+1)^2 is palindromic and prime. - Eric W. Weisstein
- A050239 Consecutive square palindromic primes; palindromic primes of the form x^2+(x+1)^2, where x are given by A050236. - Eric W. Weisstein
- A050251 Number of palindromic primes less than 10^n. - Eric W. Weisstein
- A050784 Palindromic primes containing no pair of consecutive equal digits. - Patrick De Geest
- A050786 Palindromic primes containing at least one pair of consecutive equal digits. - Patrick De Geest
- A052023 Every suffix of palindromic prime a(n), containing no '0' digit, is prime (left-truncatable). - G. L. Honaker, Jr. and Patrick De Geest
- A052024 Every suffix of palindromic prime a(n) is prime (left-truncatable). - G. L. Honaker, Jr. and Patrick De Geest
- A052025 Every prefix (and/or suffix) of palindromic prime a(n) is prime (right/left-truncatable). - G. L. Honaker, Jr. and Patrick De Geest
- A052035 Palindromic primes whose sum of squared digits is also prime. - Patrick De Geest
- A052090 Palindromic pimes (Pimes: primes whose digits contain circles). - Patrick De Geest
- A052091 Left parts needed for the construction of the palindromic prime pyramid starting with 2. - Patrick De Geest
- A052092 Lengths of the palindromic primes from Honaker's sequence A053600. - Patrick De Geest
- A052205 a(n+1) is smallest palindromic prime containing exactly 3 more digits on each end than the previous term, with a(n) as a central substring. - G. L. Honaker, Jr.
- A053054 Append n-th palindromic prime to previous term, reverse alternate terms. - Felice Russo
- A053600 a(n+1) is the smallest palindromic prime with a(n) as a central substring. - G. L. Honaker, Jr.
- A054217 Prime p concatenated with its emirp p' (prime reversal) makes a palindromic prime of the form 'primemirp' (rightmost digit of p and leftmost digit of p' are blended together - p and p' palindromic allowed). - P. De Geest
- A054218 Palindromic primes of the form 'primemirp' resulting from A054217. - Patrick De Geest
- A054797 Smallest prime number whose digits sum to nth palindromic prime. - G. L. Honaker, Jr.
- A056130 Palindromic primes in bases 2 and 4. - Robert G. Wilson v
- A056145 Palindromic primes in bases 2 and 8. - Robert G. Wilson v
- A056146 Palindromic primes in bases 4 and 8. - Robert G. Wilson v
- A056244 Palindromic primes of the form 13(n times)1. - Robert G. Wilson v
- A056245 Palindromic primes of the form 14(n times)1. - Robert G. Wilson v
- A056246 Palindromic primes of the form 15(n times)1. - Robert G. Wilson v
- A056247 Palindromic primes of the form 16(n times)1. - Robert G. Wilson v
- A056248 Palindromic primes of the form 17(n times)1. - Robert G. Wilson v
- A056249 Palindromic primes of the form 18(n times)1. - Robert G. Wilson v
- A056250 Palindromic primes of the form 19(n times)1. - Robert G. Wilson v
- A056251 Palindromic primes of the form 31(n times)1. - Robert G. Wilson v
- A056252 Palindromic primes of the form 32(n times)1. - Robert G. Wilson v
- A056253 Palindromic primes of the form 34(n times)1. - Robert G. Wilson v
- A056254 Palindromic primes of the form 35(n times)1. - Robert G. Wilson v
- A056255 Palindromic primes of the form 37(n times)1. - Robert G. Wilson v
- A056256 Palindromic primes of the form 38(n times)1. - Robert G. Wilson v
- A056257 Palindromic primes of the form 72(n times)1. - Robert G. Wilson v
- A056258 Palindromic primes of the form 74(n times)1. - Robert G. Wilson v
- A056259 Palindromic primes of the form 75(n times)1. - Robert G. Wilson v
- A056260 Palindromic primes of the form 76(n times)1. - Robert G. Wilson v
- A056262 Palindromic primes of the form 78(n times)1. - Robert G. Wilson v
- A056263 Palindromic primes of the form 79(n times)1. - Robert G. Wilson v
- A056264 Palindromic primes of the form 91(n times)1. - Robert G. Wilson v
- A056265 Palindromic primes of the form 92(n times)1. - Robert G. Wilson v
- A056266 Palindromic primes of the form 98(n times)1. - Robert G. Wilson v
- A056728 Palindromic primes using only two distinct digits and only the exterior digit is different. - Robert G. Wilson v
- A056730 Palindromic primes with just two distinct digits. - Robert G. Wilson v
- A056803 Palindromic primes of the form 12 repeated n times 1. - Robert G. Wilson v
- A057199 The first non-trivial (k>n+2) palindromic prime in both bases n and n+2. - Robert G. Wilson v
- A057332 Numbers of (2n+1)-digit palindromic primes that undulate. - Patrick De Geest
- A058375 Palindromic primes with just two distinct prime digits. - Robert G. Wilson v
- A059120 Smallest sets of 3 consecutive palindromic primes (palprimes) in arithmetic progression. The first prime of each set is listed. - Harvey Dubner
- A059121 Smallest sets of 4 consecutive palindromic primes (palprimes) in arithmetic progression. The first prime of each set is listed. - Harvey Dubner
- A059122 Smallest sets of 5 consecutive palindromic primes (palprimes) in arithmetic progression. The first prime of each set is listed. - Harvey Dubner

Jud McCranie (email) made significant progress for assignment number two - go to topic

Giovanni Resta (email) [ *May 15, 2003* ] made significant progress for assignment number one - go to topic

Zakir Seidov (email) [ *March 8, 2005* ] embedded palprimes between ones - go to topic

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E-mail address : pdg@worldofnumbers.com