[ *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));
);
}

pfgw64 "C:\pari\Mirror.txt"

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));
);
}

pfgw64 "C:\pari\MirRev.txt"

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