[ January 9, 2022 ]
Building sequences out of concatenated palindromes
Patrick De Geest

For the  ascending case  I soon discovered manually these two nice primes !

12345678911223344556677
and
123456789112233445566778899101

Is this a rare and lonely couple  77  and  101  ?
If there is a third prime the last palindrome must be larger than 10,000,000.
With each term added the numbers expands with twice the length of the last
palindrome added. Furthermore our candidate can only end with 1, 3, 7 or 9.
So you will understand why our third prime will be a very lucky shot indeed.

This is how I tackle the topic. I use Pari/gp
to make a file ('Smpal.txt') with all the candidates in it.

{
cnt=0; x=[];
for(i=1,10000000,
   p=digits(i);marque=0;ld=p[length(p)];
   if(Vecrev(p)==p, x=concat(x,p); cnt+=1; marque=1; );
   if((ld==1||ld==3||ld==7||ld==9)&&marque==1, print(cnt," ",i);
       write("C:/pari/Smpal.txt", fromdigits(x)); );
);
}

and finally check the logfile ('pfgw.log') for any (pseudo)primes.

For the  descending case  I soon discovered manually these three nice primes !

1311211111019988... ... 332211987654321

494484474464454444... ...332211987654321

747737727717707696... ... 332211987654321

This looks like a nice but isolated (?) trio  131 ,  494  and  747 .

Here is the program with adaptations for the descending case.
No need to filter out 1, 3, 7 or 9 as this kind of numbers
always end with 1. This Pari/gp will create a file ('Rsmpal.txt')

{
cnt=0; x=[];
for(i=1,10000000,
   p=digits(i);marque=0;
   if(Vecrev(p)==p, x=concat(p,x); cnt+=1; marque=1; );
   if(marque==1, print(cnt," ",i); write("C:/pari/Rsmpal.txt", fromdigits(x)););
);
}