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Old 2020-12-24, 17:51   #1
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Default Exponents of particular forms

This thread is intended as a reference thread for Mersenne exponents of special forms. For discussion, please use the reference discussion thread https://www.mersenneforum.org/showthread.php?t=23383. (Discussion posts added to this thread may be moved or deleted without notice or recourse.)

The various forms or subsets are discussed here as a sort of mathematical amusement.
There is no known basis for believing they have higher chance of revealing a Mersenne prime.
They lie at an intersection of Mersenne primes and recreational mathematics.
They might also be used as an added arbitrary selection method for sampling the natural number line segment for substantial black box QA testing of software algorithms. It's unlikely to be a good selection method compared to pseudorandom and boundary-informed selection methods.

Intro and table of contents (this post)
Mersenne rhymes https://www.mersenneforum.org/showpo...43&postcount=2
Straights https://www.mersenneforum.org/showpo...44&postcount=3
Repdigits and near-repdigits https://www.mersenneforum.org/showpo...45&postcount=4
Palindromic numbers as exponents https://www.mersenneforum.org/showpo...46&postcount=5
Prime approximations of round multiples of irrational numbers https://www.mersenneforum.org/showpo...48&postcount=6
Personally significant dates encoded into exponents https://www.mersenneforum.org/showpo...61&postcount=7
Large exponents https://www.mersenneforum.org/showpo...47&postcount=8


Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1

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Old 2020-12-24, 17:56   #2
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Default Mersenne rhymes

As used here, a rhyme is an exponent that has in common several rightmost digits with another, in the same order. Rhymes in one number base likely will not be rhymes in most others, or only to lesser extent. Except as indicated, base ten is used here. (If there is an established term for this, please PM me and I'll adopt it here.)

There is no reason to believe that if an exponent p corresponds to a Mersenne prime, that any rhyming prime exponent r = p + c x basen would also, at above the probability for a randomly chosen exponent. Yet in the category of dubious claims (or maybe math trolling), sometimes someone claims that one or more such rhyme's corresponding Mersenne number is prime. For example, 82589933, 102589933. (C 2, base 10, n 7; 7digit decimal rhyme, 0x4ec38ed, 0x61d65ed 2digit hexadecimal rhyme) emerged in this thread. Or the nearest prime to a rhyme as in https://www.mersenneforum.org/showpo...8&postcount=13, with the derivation speculatively described in https://www.mersenneforum.org/showpo...4&postcount=24.


Rhymes among exponents of known Mersenne primes
Empirically, it is straightforward to show that no two of the 51 known Mersenne primes' decimal exponents rhyme deeper than 3 decimal digits. A simple spreadsheet with exponents p, and cells p mod 10^n for n=1...4 and sorting by column for differing n is enough. Single digit rhymes are unavoidable in base 10 given the number of known Mersenne primes.
The n+1-digit rhymes are a subset of the n-digit rhymes. The expected number of n+1-digit rhymes is ~1/10 the number of n-digit rhymes for n>1. The number of rhymes for right digit 2 or 5 are zero, as for 4, 6, 8, or 0.

Rhyming decimal exponents among the known Mersenne primes, versus number of digits:
1 digit: 4 cases possible, all of which are found, 49 members of rhyme sets
There are unavoidably many matches, since, after 2 and 5, there are only 4 choices for final digit, 1, 3, 7, 9.
1 13 cases (31, 61, 521, 2281, 9941, 21701, 216091, 2976221, 20996011, 25964951, 42643801, 57885161, 74207281)
2 (2)
3 12 cases (3, 13, 2203, 4253, 4423, 11213, 86243, 110503, 859433, 6972593, 24036583, 82589933)
5 (5)
7 15 cases (7, 17, 107, 127, 607, 3217, 19937, 44497, 1257787, 3021377, 13466917, 30402457, 32582657, 37156667, 77232917)
9 9 cases (19, 89, 1279, 9689, 23209, 132049, 756839, 1398269, 43112609)
total 51 check

2 digit: 12 cases, 28 members
01 (21701, 42643801)
03 (3, 2203, 110503)
07 (7, 107, 607)
09 (23209, 43112609)
13 (13, 11213)
17 (17, 3217, 13466917, 77232917)
21 (521, 2976221)
33 (859433, 82589933)
57 (30402457, 32582657)
61 (61, 57885161)
81 (2281, 74207281)
89 (89, 9689)

3 digit: 2 cases, 4 members
281 (2281, 74207281)
917 (13466917, 77232917)

4 digit: 0 cases, 0 members; null set

2-digit rhymes are a subset of 1-digit rhymes.
3-digit are a subset of 2-digit.
4-digit would be a subset of 3-digit. Etc.
Subset of a null set is null.
5-digit and higher rhyme length are necessarily null sets.

Presumably, if the conjecture of existence of an infinite number of Mersenne primes is correct, there are exponent rhymes of any arbitrarily large length.


Testing for rhyme exponents
As a proactive measure against such guesses/trolls, I've tabulated the candidate prime rhymes in the mersenne.org exponent range 2 < r <109, and begun taking the factoring and testing of them further.


As of 2020-12-24:
Many known Mersenne primes' 7-digit decimal exponent prime rhymes for r<109 have been shown composite for all up to 100MDigit or higher:
Mp51*, Mp48*, Mp46, Mp44, Mp42, Mp41, Mp39, Mp37, Mp34, Mp31, Mp30, Mp28, Mp27, Mp26, Mp23, Mp22, Mp20, Mp19, Mp18, Mp17, Mp14, Mp13, Mp12, Mp9, Mp8, Mp7, Mp6, Mp5, Mp4, Mp3, Mp2, Mp1 (32 of the 51 known)

Of those, some are completed through 200Mdigits:
Mp5, Mp23, Mp44

Some are only one PRP test away from complete to r<109:
Mp44, Mp42, Mp39, Mp36

One is completed through r<109 (slightly higher than 300Mdigit):
Mp5 (M13)

Mp3 and Mp1 can have no prime exponents that are decimal rhymes (with 2 or 5 as right digit of the exponent)

All known Mersenne primes' 7-digit decimal exponent prime rhymes for r<109 have been P-1 factored to recommended bounds up to 100Mdigit level except:
Mp24 (260019937 is the remaining rhyme exponent, P-1 in progress)

All known Mersenne primes' 7-digit decimal exponent prime rhymes for r<109 have been trial factored to recommended bounds up to 100Mdigit exponent level.

For Mp47-Mp51* the rhyming exponents have been completed in TF and P-1 up to recommended bounds up to 200Mdigit level. Others with incomplete factoring below the 200Mdigit level:
Mp46 has a P-1 pending on one rhyme exponent to that level, as do Mp43, Mp30, Mp25 and Mp24. Mp34 and Mp27 each have a rhyme exponent on which TF is pending and P-1 is unstarted. Mp26 has a rhyme exponent on which P-1 is unstarted. Mp22 and below mostly have multiple rhymes with factoring incomplete.

TF remains to complete on >5% of identified rhymes; P-1 on >17%; PRP on >29%, of the 601 identified rhyming prime exponents < 109.


As of 2021-02-09:
Many known Mersenne primes' 7-digit decimal exponent prime rhymes for r<10 9 have been shown composite for all up to 100MDigit or higher:
Mp51*, Mp49*, Mp48*, Mp46, Mp44, Mp42, Mp41, Mp39, Mp38, Mp37, Mp34, Mp31, Mp30, Mp28, Mp27, Mp26, Mp23, Mp22, Mp21, Mp20, Mp19, Mp18, Mp17, Mp14, Mp13, Mp12, Mp9, Mp8, Mp7, Mp6, Mp5, Mp4, Mp3, Mp2, Mp1 (35 of the 51 known)

Of those, some are completed through 200Mdigits:
Mp5, Mp23, Mp44

Some are only one PRP test away from complete to r<109:
Mp44, Mp42, Mp39, Mp36, Mp18

One is completed through r<109 (slightly higher than 300Mdigit):
Mp5 (M13)

Mp3 and Mp1 can have no prime exponents that are decimal rhymes (with 2 or 5 as right digit of the exponent)

All known Mersenne primes' 7-digit decimal exponent prime rhymes for r<109 have been P-1 factored to recommended bounds up to 100Mdigit level

All known Mersenne primes' 7-digit decimal exponent prime rhymes for r<10 9 have been trial factored to recommended bounds up to 100Mdigit exponent level.

For Mp47-Mp51* the rhyming exponents have been completed in TF and P-1 up to recommended bounds up to 200Mdigit level. Others with incomplete factoring below the 200Mdigit level:
Mp46 has a P-1 pending on one rhyme exponent to that level, as do Mp25, Mp24, Mp21, Mp20, Mp16, Mp12, Mp11, Mp8, Mp7, and Mp2. Mp34 and Mp27 each have a rhyme exponent on which TF is pending and P-1 is unstarted below 200Mdigit.

TF remains to complete on <1.5% of identified rhymes; P-1 on <13%; PRP on <29%, of the 601 identified rhyming prime exponents < 109.
Estimated effort to complete remaining primality testing is about 3,900,000 GhzDays.

This state of progress is the result of many people's efforts in TF, P-1, PRP or LL as part of the overall GIMPS project.


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Old 2020-12-24, 18:00   #3
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Default Straights

These are exponents such as 160456789 which also sometimes come up in dubious claims or guesses as exponents of Mersenne primes. In the mersenne.org search space 2<p<999999999, the run of ascending digits is limited to 8 or less and not a multiple of 3. A 3-digit run segment is divisible by 3 always: c c+1 c+2 is c*100 + c*10 + c + 10 + 2 = 111c +12 which will be divisible by 3, since 111=37*3 and 12=4*3.
Both ascending straights and descending straights occur, as well as shuffled straights.
Shuffled straights would be more probable so more common than sorted straights.

Ascending straights seem to me the most interesting of the 3.

I am unaware of any effort to search for or tabulate straights that are prime exponents or the factoring or primality status of the corresponding Mersenne numbers.
There's a thread about straights as exponents.
One could generalize from straights to additional valued poker hands.


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Old 2020-12-24, 18:07   #4
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Default Repdigits and near-repdigits

Repdigits are numbers consisting only of repetition of the same digit value. (Single-digit numbers are excluded by definition.)

For Mersennes' exponents, digit values larger than one lead to composite exponents, and thereby to composite Mersenne numbers. Composite digit counts also lead to composite exponents and composite Mersenne numbers. This leaves exponents consisting of a prime number of ones, which lead to mostly composite exponents. It also leads to a very small set of candidates, since for p<109, number of digits no greater than 9, only 2, 3, 5, or 7 digits are prime length for exponent expressed in base ten.
In 1<p<999999999 Mersenne.org range:
11; prime exponent but 211-1 is divisible by 23.
111 = 3 x 37; could have ruled this out by noting sum of digits is 3
11111 = 41 x 271
1111111= 239 × 4649
So there are no Mersenne primes with base-10 repdigit exponents in 1<p<999999999 mersenne.org range.

Trying hexadecimal,
11h = 17 decimal prime exponent; 211h-1 =131071 base 10
111h = 273 = 3 x 7 x 13
11111h = 69905 = 5 × 11 × 31 × 41
1111111h = 17895697 = 29 × 43 × 113 × 127

Use of smaller number bases bring more small prime exponent lengths into range.
Near the limit, base 3
11(3) = 4 = 22
111(3) = 13 (prime) M(13) is Mp5
11111(3)= 121= 112
1111111(3) = 1093 (prime); M(1093) has 5 factors
11111111111(3) = 88573 = 23 × 3851
...7 174453 = 112 × 13 × 4561
64 570081 = 1871 × 34511

Base 2:
bits p Mp status
2 3 prime
3 7 prime
5 31 prime
7 127 prime
11 2047 = 23 × 89 so Mp is composite
13 8191 composite
17 131071 composite
19 524287 composite
23 8388607 = 47 × 178481
29 536870911 = 233 × 1103 × 2089


Near-repdigits
Exponents with the rightmost digit differing from the rest may be prime for digit values other than one repeating also.
A small perl program to find 9-digit base 10 prime exponents that are near-repdigits yields the following 7 exponents, with current status as shown:
111111113 Factored
222222227 LL DC
444444443 Factored
666666667 Tested LL & DC by LaurV
777777773 Factored
888888883 Factored
888888887 No factor to 85 bits TF completed; P-1 done; no primality test
Six composite, one remaining to be determined in the short list above.

Code:
# nearrep.pl
# perl script to find near repdigit exponents iiiiiiiij, j != i, i>0, base 10

use ntheory; 

$count=0;
for ( $i=1; $i<10; $i++ ) { #repfield is $i as digits x 8 places
  foreach $j ( 1, 3, 7, 9 ) { #rightmost
    if ($i != $j ) {
      $k=$i*11111111*10+$j;
      if ( Math::Prime::Util::GMP::is_prime($k) == 0 ) {
#        print "$k is composite\n";
      } else {
        print "$k\n";
        $count++;
      }
    }
  }
}
print "Counted $count\n(end)\n";
Requiring the differing digit to be on the right is a special case / subset of the near-repdigit definition.
Checking for other positions of the differing digit, and annotating the resulting output with current status:
Code:
# nearrep2.pl
# perl script to find near repdigit exponents i..iji..i, j != i, i>0, base 10
# where leftmost digit may be j but rightmost is not

use ntheory; 

$count=0;
for ( $l=1; $l<9; $l++ ) { #power of ten at which digit differs
  for ( $j=0; $j<10; $j++ ) { #differing digit
    foreach $i ( 1, 3, 7, 9 ) { #repfield is $i as digits x 8 places
      if ( $i != $j ) {
        $k= $i*111111111  +($j- $i) *10**$l;
        if ( Math::Prime::Util::GMP::is_prime($k) == 0 ) {
#          print "$k is composite\n";
        } else {
          print "$k\n";
          $count++;
        }
      }
    }
  }
}
print "Counted $count\n(end)\n";
Code:
333333313 factored
999999929 NF 84; P-1 NF; TF in progress
111111131 factored
111111181 LL C
777777577 factored
999999599 factored
777777677 factored
333331333 factored
999992999 NF 80, TF in progress
111113111 factored
333334333 factored
777776777 NF 85, P-1 NF, PRP in progress
777767777 factored
111181111 factored
111191111 factored
333233333 factored
999299999 NF 86, P-1 NF, PRP in progress
999499999 factored
999599999 NF 86, P-1 in progress
333733333 factored
333833333 factored
115111111 factored
776777777 factored
337333333 NF 81; P-1 NF, PRP in progress
118111111 LL C
778777777 factored
998999999 NF 86; P-1 NF, PRP in progress
101111111 factored
131111111 factored
373333333 factored
787777777 factored
577777777 factored
799999999 NF 85; P-1 NF; PRP assigned prematurely but no progress
Counted 33
(end)
23 factored, 2 LL composite primality test result, 8 remaining to be determined in the list immediately above.


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Old 2020-12-24, 18:10   #5
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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.


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Old 2020-12-24, 18:18   #6
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Default Prime approximations of round multiples of irrational numbers

Consider r = c x 10n x b, where b is an irrational number such as pi, e, sqrt(2), sqrt(3), sqrt(5), sqrt(10), etc and where c, n are are small positive integers chosen such that 108< r < 109. And find the nearest prime exponent to r. Or r = c x 10n x bm where b is a transcendental number and m is also a small positive integer.

Factoring or primality testing on such exponents may be useful by falling in regions of the number line that would not be selected otherwise in a simple QA selection method.

Some early examples of such runs revealed limitations on CUDAPm1 exponent that were later found to be partly gpu-model dependent.
141421333 (PRP C & cert) < 141421356 ~ 108 sqrt (2) < 141421387 (factored)
147576139 (factored) < 147576140 ~108 2^(1/egamma) < 147576181 (factored)
157079621 (PRP C & cert) < 157079632.7 ~ 108 pi/2 < 157079633 (factored)
161803393 (factored) < 161803399 ~108 golden ratio < 161803403 (PRP C & cert)
(all listed above <50Mdigits are completed)
173205079 (PRP in progress) < 173205081 ~ 108 sqrt (3) < 173205089 (factored)
223606793 (PRP in progress) < 223606798 ~ 108 sqrt (5) < 223606807 (PRP in progress)
271828171 (PRP in progress) < 271828183 ~108 e < 271828199 (PRP C, cert pending)
292005097 (factored) < 292005098 ~ 108 "Buenos Aires constant" < 292005113 (PRP assigned & queued)
314159257 (LL assignment stagnant; PRP in progress) < 314159265 ~108 pi < 314159311 (factored)
316227731 (factored) < 316227766 ~ 108 sqrt(10) < 316227767 (factored)
(As of 2021-01-29, all listed above that are >50Mdigits & <100Mdigits are factored, or completed in TF and P-1 to recommended bit levels and bounds and primality tests are assigned)
543656363 (factored) < 543656366 ~2 x 108 e < 543656371 (TF, P-1 done, PRP in progress)
628318517 (TF & P-1 done) < 628318531 ~2 x 108 pi < 628318583 (TF & P-1 done)
(As of 2021-01-29, all listed above that are >100Mdigits & <200Mdigits are factored, or completed in TF and P-1 to recommended bit levels and bounds)
(As of 2021-02-27, all listed below >200Mdigits are factored, or TF completed to recommended bit levels and P-1 done to at least recommended bounds)
785398129 (factored) < 785398163 ~ 109 pi/4 < 785398169 (factored)
853973387 (factored) < 853973422 ~ 108 e pi < 853973437 (TF, P-1 done)
942477787 (factored) < 942477796 ~3 x 108 pi < 942477799 (TF done, P-1 done)
986960431 (factored) < 986960440 ~108 pi2 < 986960461 (TF & P-1 done)
Suggestions for additions to the above list are welcome by PM.


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Old 2020-12-24, 21:23   #7
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Default Personally significant dates encoded into exponents

The idea of encoding personally significant dates into exponents came up in https://www.mersenneforum.org/showpo...04&postcount=8 and has no more mathematical merit than trying to pick a winning lottery ticket by the same method, which is no merit at all. But if it's fun, and you're not concerned about the leakage of personally identifying information, try it. Birthdates, anniversaries, etc.


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Old 2021-01-22, 16:45   #8
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Default Large exponents

Here, large is defined as an exponent beyond the mersenne.org limit, currently 109, which is almost 230
Before considering larger exponents, it's useful to consider the scale of effort needed near the limit of mersenne.org.

TF to recommended bit levels for near the 109 limit takes days per exponent on the fastest gpus and best software.
P-1 testing to recommended bounds near the 109 limit takes ~4 days per exponent in GpuOwL on fast gpus such as Radeon VIIs.
Primality testing near the 109 limit takes around 6 months per exponent with GpuOwL on a Radeon VII. Prime95 benchmarks indicate about 13 months each primality test near 109 on a Xeon Phi 7250 (68 cores 1 worker at ~1.4-1.5 GHz clock and 16GB 7200MHz MCDRAM in same chip package).

ONLY PRP should be considered for such long tests. (LL testing lacks the Gerbicz error check in all existing software and lacks even the weak Jacobi symbol check in some software, so is unlikely to complete correctly for very long runs.)

Primality testing effort scales as roughly exponent2.1. P-1 effort and TF effort scale similarly, with a smaller proportionality constant, ~1/40 that of primality testing, each. Conversely, the probability of finding a prime diminishes per test as exponent increases.

Software support for large exponents is limited, and the more so as exponent increases. There is little reason to exert effort in software development for large exponents soon, since the remaining work in mersenne.org will take more than a century at currently projected rates. https://www.mersenneforum.org/showpo...5&postcount=11

A summary of software support is available in the attachment at http://www.mersenneforum.org/showpos...91&postcount=2
Interim residues for a limited wide ranging variety of fft lengths and specified exponents are available at https://www.mersenneforum.org/showpo...4&postcount=12

https://www.mersenne.ca/ covers status for exponents up to 1010, and coordinates trial factoring for exponents over 109 up to 232.

Gigadigit Mersennes are covered in subforum Operation Billion Digits https://mersenneforum.org/forumdisplay.php?f=50.
Assignment reservation and result submission page, and display of status for trial factoring is at https://www.mersenne.ca/obd
I do not know of any software currently capable of P-1 factoring such large numbers in a reasonable amount of time or with acceptable reliability. See https://mersenneforum.org/showthread.php?t=24486 for discussion of run time scaling and requirements. It's likely Mlucas will have sufficient P-1 capability implemented as a byproduct of the coming F33 attempt.
Gigadigit primality tests are technically possible now in Mlucas or certain versions of Gpuowl, but the duration per test on available hardware is longer than the likely hardware lifetime.So the hardware would need to be replaced and work in progress migrated, and great care exerted with backups of work in progress and error checking along the way.

There's a forum thread regarding 10-gigadigit numbers, in which even larger are also mentioned. Only trial factoring is feasible for these. There's little or no point in doing so, other than as an amusement. https://mersenneforum.org/showthread.php?t=16801

Double Mersennes: For Mersenne numbers with Mersenne numbers as exponents, see the Operazione Doppi Mersennes subforum: https://mersenneforum.org/forumdisplay.php?f=99. Also http://www.doublemersennes.org/ There's a thread discussing when it might be feasible to primality test MM61. Assuming Moore's law continues, over 100 years. But Moore's law is slowing and faces hard physics limits. https://mersenneforum.org/showthread.php?t=21522
Some of the challenges and changes required are discussed briefly in https://mersenneforum.org/showthread.php?t=17354 and https://mersenneforum.org/showpost.p...6&postcount=16

There's also a bit of discussion of the difficulty of even attempting TF on the triple Mersenne MMM127. https://mersenneforum.org/showthread.php?t=18522


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