xddd

[sweety439]

Update files for unique primes

[sweety439]

format of file:

b,x,{y}: smallest prime of the form xyyy...yyy in base b
b,{x},y: smallest prime of the form xxx...xxxy in base b

[sweety439]

A305547
A305549
A305550
A326611
A326613
A326616
A326617
A333246
A333247
A333322
A333323
A333324
A333325
A333326
A333327
A333328
A333329
A333330
A333331
A333332
A333333
A333334
A333335
A333336

[tuckerkao]

Thanks a lot sweety439 for providing me the link to Wiki that shows the known Mersenne Primes in the dozenal base, looks like the dozenal 7 enders come mostly with 77 or 87 for the larger size exponents.
https://dozenal.fandom.com/wiki/Mersenne_prime

Thus, I've created a list, so I can run the trial factoring tasks for those in higher priority -

Mersenne Exponent # in Decimal -> Dozenal (without known factors)

M168005623 -> Z48321587
M168030523 -> Z48333Ӿ77
M168056023 -> Z48346787
M168081223 -> Z48359287
M168084823 -> Z4835Ɛ387
M168088423 -> Z48361487

M168149623 -> Z48390987
M168156523 -> Z48394977
M168160123 -> Z48396Ӿ77
M168181723 -> Z483Ӿ7477
M168182023 -> Z483Ӿ7687
M168189223 -> Z483ӾƐ887
M168196123 -> Z483Ɛ3877

M168207223 -> Z483ƐӾ187
M168243223 -> Z48416Ɛ87

M168333223 -> Z4845Ɛ087
M168365623 -> Z48475987
M168394423 -> Z4848Ӿ587
M168398023 -> Z48490687

M168433723 -> Z484Ӿ9277 (I'm running the PRP test for this one)
M168437623 -> Z484ӾƐ587
M168462823 -> Z48502087
M168470023 -> Z48506287
M168477223 -> Z4850Ӿ487
M168484123 -> Z48512477
M168494923 -> Z48518777

M168549223 -> Z48544087
M168592423 -> Z48565087
M168599323 -> Z48569077
M168599623 -> Z48569287

M168624823 -> Z4857Ɛ987

M168757723 -> Z48624877
M168779323 -> Z48635277
M168797623 -> Z48643987

M168891223 -> Z48689Ɛ87
M168898123 -> Z48691Ɛ77

M168930823 -> Z486Ӿ8Ӿ87
M168934123 -> Z486ӾӾ977
M168945223 -> Z486Ɛ5287
M168973723 -> Z48709877
M168998923 -> Z48720377

[tuckerkao]

Similar ending digits are observable from both the decimal and dozenal exponents.

M82589917 -> Z237ӾƐ111
M168433717 -> Z484Ӿ9271

I'll factor up the larger exponent, see whether it can survive the trial factoring and P-1 to the recommended level and bounds.

[sweety439]

Originally Posted by tuckerkao

Similar ending digits are observable from both the decimal and dozenal exponents.

M82589917 -> Z237ӾƐ111
M168433717 -> Z484Ӿ9271

I'll factor up the larger exponent, see whether it can survive the trial factoring and P-1 to the recommended level and bounds.

The number of Mersenne exponents end with E should be less than that of Mersenne exponents end with 1, 5, or 7, this is because for all Sophie Germain primes p == 3 mod 4 (all such primes end with E), Mp is divisible by 2*p+1, thus Mp is composite.

The Fermat primes and Mersenne primes in dozenal also have interesting properties:

* Except for 3, all Fermat primes end with 5. (In fact, there are only 5 known Fermat primes (3, 5, 15, 195 and 31E15) and it is conjectured that there are no more Fermat primes, interestingly, all digits of all known Fermat primes are odd)

* Except for 3, all Mersenne primes end with 7. (Besides, all Mersenne primes except 3 and 7 end with one of the only two 2-digit Mersenne primes (27 and X7))

[Dr Sardonicus]

Originally Posted by sweety439

The number of Mersenne exponents end with E should be less than that of Mersenne exponents end with 1, 5, or 7, this is because for all Sophie Germain primes p == 3 mod 4 (all such primes end with E), Mp is divisible by 2*p+1, thus Mp is composite.
<snip>

I'm not sure why "E" is here used to denote decimal eleven - possibly because it's the first letter of the word "eleven?" In any case, it's gratuitously confusing. In the widely-used standard notation for hexadecimal, the first letters of the alphabet, A through F, are used in order to denote the hex digits greater than nine. Using A for decimal ten and B for decimal eleven in base twelve would be consistent with this usage.

Of the 49 exponents p > 3 of Mersenne primes, the number of p%12 = 1,5,7,11 are 11, 20, 13, and 5 respectively.

I would question any significance, either of the relative scarcity of exponents p == 11 (mod 12), or the relative plenty of those congruent to 5 (mod 12) because the total number of exponents is small.

I would qualify the significance of Sophie Germain primes 4n + 3, 8n + 7 as applying only to "small" exponents. The best guess about the density of Sophie Germain primes is, the number of them less than X is asymptotically c*X/log²(X), hence an infinitesimal proportion of all primes.

[LaurV]

I agree with the fact that using E for eleven is a bit forced, and somehow confusing (with 3, or 8, or with E=14 in hex). On the other hand, I never liked using B or D in the hex system (they are easily confused with other characters when displayed on low-cost LCDs, like 7-segments or so, and B and 8 are too similar in hex strings (like coin addresses or hash tables) and difficult to read if the font is chosen in an "unfortunate" way. As well as T (for ten) which is can be taken as 1 or 7. Many other "better" solutions were suggested, but all have their drawbacks. The "A to F" has the advantage that is easier "transformable" (symbols are consecutive, as opposite of other ideas that would use H and L for example, these are readable on 7-segments, and are ready available in some industrial systems to show High/Low levels of voltage, pressure, whatever). The "best way" in my opinion, would be to use new symbols (unicode has a section), to have them in a contiguous way, from, say, zero to 60 (larger bases won't make much sense), but again, A to F are ready-made, and single-byte available in the ASCII code. Don't fix the system that it works!

[sweety439]

Originally Posted by LaurV

I agree with the fact that using E for eleven is a bit forced, and somehow confusing (with 3, or 8,
..............................................................................

If I only research a problem in dozenal (base 12), or if I only research a problem in bases <= 12, then I use X for ten and E for eleven, however, if I research a problem in bases including those > 12, then I use A for 10, B for 11, C for 12, ..., Z for 35

[Uncwilly]

There is no need to quote the entire immediately preceding post when replying.

[tuckerkao]

Originally Posted by LaurV

A to F are ready-made, and single-byte available in the ASCII code. Don't fix the system that it works!

The only time it won't work is to use the dozenal and music systems at the same time. A major and A minor are not 5 * 2, so they cannot be assigned with the same symbol on the same equation for 2 distinctive purposes.

It starts with the D key because D is the only white piano key that is symmetrical in the real shape, thus important for the mirrored patterns to stand out between the right and the left hand fingerings.


Right Hand:

D = 0
D#/E♭ = 1
E = 2
F = 3
F#/G♭ = 4
G = 5
G#/A♭ = 6
A = 7
A#/B♭ = 8
B = 9
C = Ӿ
C#/D♭ = Ɛ
D = 10


Left Hand:

D = 0
C#/D♭ = -1
C = -2
B = -3
A#/B♭ = -4
A = -5
G#/A♭ = -6
G = -7
F#/G♭ = -8
F = -9
E = -Ӿ
D#/E♭ = -Ɛ
D = -10


Piano keyboard is something that cannot be decimalized, not possible to take 2 keys out an octave. Ɛ is the symbol the dozenal society uses not E. Using Ӿ instead of X to avoid the similarities from the multiplication sign.