Palindromic numbers are numbers that are reversible digit by digit without changing value.

110505011 is an example, which as a Mersenne exponent, leads to a P-1 factor.

Palindromic numbers in base ten are quite common. The subset that are primes are also common. There are 5172 in 10

^{8}<p<10

^{9}. As of 2021 Jan 19, over 3100 of those have been factored, and more than 200 others have PRP or LL composite primality test results.

Palindromic number exponents 10 < p < 10

^{8} have already been tested at least once by GIMPS and indicated composite. None of the currently known Mersenne primes have an exponent that is a palindromic number of 2 or more digits in base ten. By definition, single digit numbers are palindromic, so the 4 known Mersenne primes that have palindromic numbers as exponents in base ten are M(2), M(3), M(5), M(7).

A subset of palindromic numbers contain shorter palindromic numbers. For example,

171575171 which is a prime exponent and the corresponding Mersenne number has a minimal factor. f=2kp+1 = 34315

**0**343, k=1

https://www.mersenne.ca/exponent/171575171. That factor is nearly palindromic.

Or those of form

p*10

^{6}+(2p+1)*10

^{3}+p,

such as

171343171, which also is a prime exponent and the corresponding Mersenne number has a known factor. There are also some palindromic exponents with embedded palindromic prime numbers. An example of base ten 9-digit palindromic prime numbers of form

a (10

^{6}+1)+b 10

^{3}
containing only 3-digit palindromic prime numbers a and b, of form

c (10

^{2}+1) + 10 d

containing only 1-digit primes, is 373 353 373.

373353373
Please PM Kriesel with what those special cases are called.

Top of reference tree:

https://www.mersenneforum.org/showpo...22&postcount=1