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 a1call 2022-02-04 23:12

[url]https://www.wolframalpha.com/input?i=48532837_12+to+base+10[/url]

[url]https://www.wolframalpha.com/input?i=168526123+is+prime[/url]

[url]https://www.mersenne.org/report_exponent/?exp_lo=168526123&exp_hi=[/url]

 paulunderwood 2022-02-04 23:19

:redface: I see. Convert to base 10 from base 12. Silly me. Got for it Tucker! No one is stopping you. Good luck!

(I initially thought that T. was saying that converting a prime to another base somehow changed its primeness. Sorry).

 Dobri 2022-02-05 00:16

[QUOTE=tuckerkao;599408]So I've decided that I want to run a PRP test of 2[SUP]48,532,837[/SUP] - 1 myself(This exponent is a decimal composite but a dozenal prime with a different interpretation),...[/QUOTE]
Note that 48,532,837 = 2,281 × 21,277 and two of the factors of (2[SUP]48532837[/SUP] - 1) are (2[SUP]2281[/SUP] - 1) and (2[SUP]21277[/SUP] - 1) where (2[SUP]2281[/SUP] - 1) is the 17th Mersenne prime M[M]2281[/M] and M[M]21277[/M] has a known factor.

 Dobri 2022-02-05 00:53

[QUOTE=tuckerkao;599408]So I've decided that I want to run a PRP test of 2[SUP]48,532,837[/SUP] - 1 myself(This exponent is a decimal composite but a dozenal prime with a different interpretation),...[/QUOTE]
As for 48,532,837[SUB]12[/SUB] = 168,526,123[SUB]10[/SUB], it is a prime number indeed.
Off-topic: [SPOILER]Throughout the Chinese Lunar Year Huangdi (Yellow Emperor) 4720 (2022) of the Water Tiger, lucky numbers allegedly are 1, 3, and 4 as well as numerals that contain them such as 14 and 34. The digit sum of 168,526,123 is 34 and there is no known Mersenne prime having an exponent with said digit sum. :)[/SPOILER]

 sweety439 2022-02-08 05:17

Other than single-digit primes (2, 3, 5, 7), can a Mersenne exponent be a [URL="https://en.wikipedia.org/wiki/Palindromic_prime"]palindromic prime[/URL]?

 kriesel 2022-02-08 19:09

[QUOTE=sweety439;599635]Other than single-digit primes (2, 3, 5, 7), can a Mersenne exponent be a [URL="https://en.wikipedia.org/wiki/Palindromic_prime"]palindromic prime[/URL]?[/QUOTE]For Mersenne numbers, there are ~5172 prime palindromic exponents of 9 decimal digits. Many of them have already been factored or otherwise shown composite. All the shorter palindromic exponent Mersennes have already been tested and found composite. There are no known Mersenne primes with exponent > 10 and palindromic exponent. Primality testing passed above 10[SUP]8[/SUP] a while ago. But the sample size of known Mersenne primes' exponents is small at 51. If the exponent distribution of Mersenne primes is random in some sense, the odds of any of the estimated 6 Mersenne primes remaining to be discovered below 1G exponent having a palindromic exponent are very poor, at ~6 chances, each with odds ~1/4000 of having a palindromic exponent, or roughly 6/4000 overall. p=abcd e dcba would not allow a=2,4,5,6,8, only a=1,3,7,9.
So we already know there are no (base 10) palindromic exponent Mersenne primes between exponents:
11. - 60M & probably to 107M; 200M-300M; 400M-700M; 800M-900M;
leaving 60M likely 107M - 200M, 300M - 400M, 700M - 800M, 900M+.

The odds get worse for 10 or more digit exponents.
[URL]https://www.mersenneforum.org/showpost.php?p=567246&postcount=5[/URL]

Changing bases, there are more. Consider base two. 3, 5, 7, 17, 31, 107, 127.

 chalsall 2022-02-08 19:14

[QUOTE=kriesel;599666]The odds get worse for 10 or more digit exponents.[/QUOTE]

I know it's not easy to do...

But... I keep being told by those I trust that speaking into a vacuum means (by definition) that few actually receive your message.

 Dr Sardonicus 2022-02-08 20:36

[QUOTE=kriesel;599666]<snip>
The odds get worse for 10 or more digit exponents.
<snip>[/QUOTE]If k > 1, the odds of a palindromic number with 2*k digits being prime are [i]zero[/i]. (This is true in any integer base b > 1; the reason is that such numbers are automatically divisible by b + 1, with cofactor automatically greater than 1 if k > 1.)

For positive integer k, the number of 2*k + 1 digit palindromic numbers to base ten is 9*10[sup]k[/sup] (9 possible non-zero digits for the first and last digit, ten possible middle digits, and an arbitrary block of k-1 digits in between).

Of these, 4*10[sup]k[/sup] are relatively prime to 10. Apart from that, I have no idea of the likelihood of a palindromic number of 2*k + 1 decimal digits being prime. Under the "assumption of ignorance" that the likelihood is the same as a random odd number prime to 10, something on the order of 1/(2*k*log(10)) of them would be prime. But the total number of palindromic numbers is tiny compared to the number of primes if k is large.

The number of primes with 2*k + 1 decimal digits is roughly 9*10[sup]2k[/sup]/((2k)*log(10)), so the odds of a 2k+ 1 digit prime being palindromic are less than 8*k*log(10)/9 in 10[sup]k[/sup]. Under the "assumption of ignorance" the odds would be something like 4/9 in 10[sup]k[/sup] (unless I botched the calculation, of course) :grin:

 sweety439 2022-03-21 18:39

R49081 is now proven prime, see post [URL="https://mersenneforum.org/showpost.php?p=602219&postcount=35"]https://mersenneforum.org/showpost.php?p=602219&postcount=35[/URL]

 sweety439 2022-03-22 11:33

[QUOTE=Dr Sardonicus;602253]The University of Illinois used to have a post mark commemorating its achievement finding the latest Mersenne prime, proving that

[color=red]2[sup]11213[/sup] - 1 IS PRIME[/color]

Imagine the possibilities of putting

[b][i][size=+1]R[/size][/i][/b][sub]49081[/sub] IS PRIME

on something :big grin:[/QUOTE]

2^11213-1 can be easily proven prime because its N+1 can be trivially 100% factored:

Pocklington [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 primality test[/URL] --> [URL="https://en.wikipedia.org/wiki/Proth%27s_theorem"]Proth primality test[/URL] --> [URL="https://en.wikipedia.org/wiki/P%C3%A9pin%27s_test"]Pépin primality test[/URL] for Fermat numbers

Morrison [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 primality test[/URL] --> [URL="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer%E2%80%93Riesel_test"]Lucas–Lehmer–Riesel primality test[/URL] --> [URL="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test"]Lucas–Lehmer primality test[/URL] for Mersenne numbers

However, for R49081, neither N-1 nor N+1 can be trivially >= 33.3333% factored, thus ECPP primality test (such PRIMO) is needed to use, thus they are very different.

See [URL="https://primes.utm.edu/primes/lists/all.txt"]top definitely primes[/URL] and [URL="http://www.primenumbers.net/prptop/prptop.php"]top probable primes[/URL], for the top definitely primes, (usually) one of N-1 and N+1 is trivially 100% factored, while for top probable primes, none of them can be >= 33.3333% factored.

 rudy235 2022-03-22 15:05

[QUOTE=sweety439;602283]2^11213-1 can be easily proven prime because its N+1 can be trivially 100% factored:[/QUOTE]

2[SUP]11213[/SUP]-1 would _not_ use BLS or CHG for N-1 as in this case LL is quicker. On the other hand[B] R[/B][SUB]1031[/SUB] was proven by a procedure similar to BLS [FONT="Fixedsys"]Brillhart-Lehmer-Selfridge[/FONT]

See [URL="https://www.ams.org/journals/mcom/1986-47-176/S0025-5718-1986-0856714-3/S0025-5718-1986-0856714-3.pdf"]https://www.ams.org/journals/mcom/1986-47-176/S0025-5718-1986-0856714-3/S0025-5718-1986-0856714-3.pdf[/URL]

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