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Old 2020-12-24, 18:10   #5
kriesel's Avatar
Mar 2017
US midwest

712 Posts
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 That factor is nearly palindromic.
Or those of form
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.

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