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Old 2020-12-24, 18:10   #5
kriesel
 
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Mar 2017
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Default Palindromic numbers as exponents

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 108<p<109. 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 < 108 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 = 343150343, k=1 https://www.mersenne.ca/exponent/171575171. That factor is nearly palindromic.
Or those of form
p*106+(2p+1)*103+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 (106+1)+b 103
containing only 3-digit palindromic prime numbers a and b, of form
c (102+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

Last fiddled with by kriesel on 2021-02-28 at 20:15 Reason: add subset examples
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