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Old 2020-10-10, 18:12   #12
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Default Mersenne exponents of particular forms

Thread with intro, separate posts re forms

The various forms or subsets are discussed here as a sort of mathematical amusement. They might also be used as an 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.


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.

There is no reason to believe that if an exponent p corresponds to a Mersenne prime, that any p+c*basen would also, at above the probability for a randomly chosen exponent. Yet in the category of dubious claims, people often speculate that one or more such rhymes corresponding Mersenne number are prime. For example, 82589933, 102589933. (C 2, base 10, n 7; 7digit decimal rhyme, 0x4ec38ed, 0x61d65ed 2digit hexadecimal rhyme)

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 knowns.
n+1-digit rhymes are a subset of 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, 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 (31, 61, 521, 2281, 9941, 21701, 216091, 2976221, 20996011, 25964951, 42643801, 57885161, 74207281
2 (2)
3 12 (3, 13, 2203, 4253, 4423, 11213, 86243, 110503, 859433, 6972593, 24036583, 82589933)
5 (5)
7 15 (7, 17, 107, 127, 607, 3217, 19937, 44497, 1257787, 3021377, 13466917, 30402457, 32582657, 37156667, 77232917)
9 9 (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.




Straights
These are exponents such as 160456789. In 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.



Repdigits
Repdigits are numbers consisting of repetition of the same digit value.

For Mersennes, 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.
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


Palindromes
Palindromes 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.
Palindromes in base 10 are quite numerous. The subset that are primes are also numerous. There are 5170 in 108<p<109.


Prime approximations of round multiples of irrational numbers

(examples)

Last fiddled with by kriesel on 2020-10-10 at 22:48
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Old 2020-10-12, 22:18   #13
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Something about ecm memory and time scaling in prime95
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Old 2020-10-24, 20:42   #14
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GPU models no longer available to me for test
GTX 480
Tesla C2075
GTX 1070
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