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 base

^{n} 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 <10

^{9}, 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<10

^{9} 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<10

^{9}:

Mp44, Mp42, Mp39, Mp36

One is completed through r<10

^{9} (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<10

^{9} 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<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 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 < 10

^{9}.

**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<10

^{9}:

Mp44, Mp42, Mp39, Mp36, Mp18

One is completed through r<10

^{9} (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<10

^{9} 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 < 10

^{9}.

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.

Top of reference tree:

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