[ November 12, 2002 ]
Palindromic Primes stretching out until...
Jeff Heleen (email)

I have found 3 palprimes. I don't know if you have them
somewhere already on your site but here they are.
If you start with a central 1 and concatenate the
counting integers to either side (mirrorwise to the left),
the palprimes occur at N = 31, 59 and 113,
all three N values are coincidentally prime !

1303928272...7654321234567...2728293031 = prime (105 digits)
9585756555...7654321234567...5556575859 = prime (217 digits)
311211111011...4321234...110111112113 = prime (417 461 digits)

My search went up to N = 999. I used Primo for the tests.

Who can extend this list with a few more terms ?

[ November 15, 2002 ]
Jean Claude Rosa wrote that the third number
had a wrong length indication 417 instead of 461.
"Soit P = n... ...32123... ...n avec 99 < n < 1000
pour avoir la longueur de P j'utilise la formule suivante :
longueur de P = 6 * n - 217
Si n = 113, longueur de P = 6 * 113 - 217 = 461."

[ November 17, 2002 ]
Jeff Heleen has another possible palprime for this plate.
For N = 1277 (a prime itself) shows promise. This would yield
a number with 8001 digits (assuming I have added correctly).
I shall not attempt to prove it prime at this time as it
would take far too much time.

PDG tested this candidate with PFGW [ July 13, 2004 ]
but to my surprise the outcome was that this number is composite !
In case I made a mistake perhaps someone would like
to confirm that this palindrome is not a probable prime ?

[ February 3, 2022 ]
I, Patrick De Geest, refreshed the above search but now
with the aid of a Pari/gp script. All it does is to make a list
of all the candidate palindrome primes into a file "Mirror.txt".
From there on that list is passed through to pfgw64 who tests
all those entries for probable-prime-ness. Here is my code.

{
x=[];y=[];
for(i=1,19999,
   p=digits(i); x=concat(x,p); y=(Vecrev(x));
   z=fromdigits(concat(y, x[2..length(x)])); ld=x[length(x)];
   if((ld==1||ld==3||ld==7||ld==9), print(i," ",z); write("C:/pari/Mirror.txt", z));
);
}

In the Pfgw64 folder you'll find a log file (pfgw.log)
with all the 3-PRP!'s he found.
Note that the candidate with N = 9999 has 77777 digits !

The mirrorleft and right number N reached now 19999 (177,777 digits) [ March 18, 2022 ]
and still no fourth (probable) prime.

So, if there is a fourth companion N must be greater than
 31   59   113   N > 19999 

There is of course the similar case where we have as left part
the concatenations of the descending numbers (without reversing them)
or the Rsm series 13_12_11_10_9_8_7_6_5_4_3_2_1
followed on the right by the mirror image of the above Rsm section
13_12_11_10_9_8_7_6_5_4_3_2_1_2_3_4_5_6_7_8_9_01_11_21_31

{
x=[];y=[];
for(i=1,19999,
   p=digits(i); x=concat(p,x); y=(Vecrev(x));
   z=fromdigits(concat(x, y[2..length(y)])); ld=y[length(y)];
   if((ld==1||ld==3||ld==7||ld==9), print(i," ",z); write("C:/pari/MirRev.txt", z));
);
}

Two primes popped up after a short run of the Pari/gp code.

[ 7372717069... ...12111098765432123456789011121... ...6907172737 ] with 273 digits
and
[ 130129128127 ... ...1110987654321234567890111... ...721821921031 ] with 563 digits

Let me now go throught the higher values and see if more primes shows up.

The mirrorright and left number N reached now 19999 (177,777 digits) [ May 8, 2022 ]
and still no third (probable) prime.

For now I collected  73   130   N > 19999