[ 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 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).
1. Two 1-d palprimes inserted in 11 giving palprimes are
3 and 5.
So 131 and 151 are palprimes.
2. Three 3-d palprimes inserted in 11 giving palprimes are
131, 383, 797.
So 11311, 13831, 17971 are palprimes.
3. Nine 5-d palprimes inserted in 11 giving palprimes are
11411, 16061, 16361,
19391, 33533, 36263,
73037, 75557, 79397.
4. Ninety three 7-d palprimes inserted in 11 giving palprimes are
5. 421 are 9-digit palprimes (from a total of 5172) which when inserted in 11 give 1pp1 palprimes. See also A088269, A103992.
.
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.
Assignments
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.
[ 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 :
Here is just one solution (303 digits) from his email that
I prefer above the others because of its palindromic length !
p = 10302 + 10*(7*10301-70)/99 + 10140*22001012200100221010022 + 1
Let w be "0" followed by 69 concatenations of "70".
Then p is the concatenation 1w92708082907170928080729w1
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 (!)).
%N Emirps (primes whose reversal is a different prime). under A006567.
Click here to view some of the author's [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.
Albert H. Beiler wrote in his book "Recreations in the Theory of Numbers" Second Edition 1966 page 222 :
Primes can even be palindromic (reading the same backward as forward) and be in arithmetic progression.
The respective common differences are 810; 3030; 3030 and 300.
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."
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."
More Integer Sequences from Sloane's OEIS Database
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
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
Contributions
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
Palindromic Primes 5 Page 5
1 
Is the number 10501 a unique palindromic prime ?Solution equation: (p-138) + (p) + (p+158) = (3p+20)
The Top Ten by Rudolph Ondrejka
Here is an excerpt from Martin Gardner's book
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