Are all MP# produced by logarithmic numbers beginning with 2 or 8 in ASCII?

[WilliamJohnCox]

In completing a paper for publication, Calculating the Geometry and Numbers of Universal Mathematics, which inserts the UNStar symbols for the powers of two in a alternative apt for ASCII: 1,2,3,U – 4,5,6,U – 7,8,9,S – C,X,W,10.

I just completed a table of the proportions of the Perfect Prime powers of two which produce the Perfect Logarithmic Numbers, which, when reduced by one, produce the Mersenne Prime numbers, which when calculated produce the Perfect Numbers. In UN, these Perfect Primes run: 2,3,4,6,C,11,13,1W,3C,47,59,6W,207,24W,UWW,N79,NXY,S71, etc for the full list of 33 (51).

I am using these seminal prime numbers as computational proxies for their progeny (Mersenne Prime and Perfect Numbers) to calculate their proportions. So far as I could determine, given my limited computing capacity, these UN Perfect Prime numbers result in Perfect Logarithmic Numbers producing a Mersenne Prime: U, N, 20, N0, 2:000, 20:000, N0:000.

My question is: are all known Mersenne Primes produced by Perfect Logarithmic Numbers, and if so, do they all begin with a 2 or N in UN, or 2 or 8 in ASCII? If so, would a concentration on these two numbers be of any value in narrowing the search for Mersenne Primes? Or, or is GIMPS already doing this? Might there be some relationship to the fact that in base 10, some perfect numbers end in 6s and others in 8s. UN Perfect Numbers organize as Ws and 0s, occasionally separated by an S in the same number as the one in base 10 ending in eight. (For example, 1WWWWS0000 or 137,438,691,328 in base 10).

For interest, a python UN basic calculator is available, and my 30-page Introduction to Universal Quantum Numbers can be accessed at https://williamjohncox.com/UNIntro.pdf.

Thanks,
~wm

[retina]

Originally Posted by WilliamJohnCox

... symbols for the powers of two in a alternative apt for ASCII:[I] 1,2,3,U – 4,5,6,U – 7,8,9,S – C,X,W,10.

Why all the unnecessary complication? Just use ordinary base-10 or hex and then people will understand you better.

Plus, the ambiguous repeated symbol "U" makes it all look very silly.

[kriesel]

Originally Posted by retina

Why all the unnecessary complication? Just use ordinary base-10 or hex and then people will understand you better.

Perhaps clarity or accessibility of checking for correctness is not the point.

From the linked pdf:

"decimal base-10 was hyped to hexadecimal ASCII by
inserting A, B, C, D, E, and F as numbers between nine and ten"
Um no. "A" represents one more than 9 in hexadecimal, which would be ten. A one followed by a zero represents sixteen in hexadecimal, not ten.

"Two years ago, after 20 years of searching, a volunteer identified the 51st Mersenne Prime, 2^82,589,933" (which is actually extremely composite).
It's not true that the lucky finder of 2^82589933-1 has been at it for 20 years.
"After less than 4 months and on just his fourth try, he discovered the new prime number." https://www.mersenne.org/primes/?press=M82589933

in my opinion Cox gets 10 points for #4 of https://primes.utm.edu/notes/crackpot.html

There's much more.

There seems to me a substantial vanity self-publishing component.

[Dobri]

Originally Posted by WilliamJohnCox

My question is: are all known Mersenne Primes produced by Perfect Logarithmic Numbers, and if so, do they all begin with a 2 or N in UN, or 2 or 8 in ASCII? If so, would a concentration on these two numbers be of any value in narrowing the search for Mersenne Primes?

Yes, the first digit of 2^MersennePrimeExponent in the base-16 numeral system is always 2 or 8 except for M(1) which is 2^2 = 4.
This is of no value for narrowing the search for Mersenne primes except for the empirical observation that the first digit is more often 2 than 8 in the small sample of known Mersenne primes.

The following simple code written in Wolfram language (Wolfram Mathematica is distributed for free on Raspberry Pi devices) shows the distribution of 2 and 8 as first digits of 2^MersennePrimeExponent in the base-16 numeral system for the known Mersenne primes.

Mexponent = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
43112609, 57885161, 74207281, 77232917, 82589933};
n = 1; While[n <= 51, x = 2^Mexponent[[n]]; Print[n, " ", Mexponent[[n]], " ", NthDigit[x, 1, 16]]; n++];

#, Mexponent, First digit of 2^Mexponent in the base-16 numeral system
1, 2, 4
2, 3, 8
3, 5, 2
4, 7, 8
5, 13, 2
6, 17, 2
7, 19, 8
8, 31, 8
9, 61, 2
10, 89, 2
11, 107, 8
12, 127, 8
13, 521, 2
14, 607, 8
15, 1279, 8
16, 2203, 8
17, 2281, 2
18, 3217, 2
19, 4253, 2
20, 4423, 8
21, 9689, 2
22, 9941, 2
23, 11213, 2
24, 19937, 2
25, 21701, 2
26, 23209, 2
27, 44497, 2
28, 86243, 8
29, 110503, 8
30, 132049, 2
31, 216091, 8
32, 756839, 8
33, 859433, 2
34, 1257787, 8
35, 1398269, 2
36, 2976221, 2
37, 3021377, 2
38, 6972593, 2
39, 13466917, 2
40, 20996011, 8
41, 24036583, 8
42, 25964951, 8
43, 30402457, 2
44, 32582657, 2
45, 37156667, 8
46, 42643801, 2
47, 43112609, 2
48, 57885161, 2
49, 74207281, 2
50, 77232917, 2
51, 82589933, 2

[tuckerkao]

Hi Dobri, I can do the hex-base pretty well. The most memorable Hex Mersenne Prime is ADD1₁₆.

1. 2
2. 3
3. 5
4. 7
5. D
6. 11₁₆
7. 13₁₆
8. 1F₁₆
9. 3D₁₆
A. 59₁₆
B. 6B₁₆
C. 7F₁₆

D. 209₁₆
E. 25F₁₆
F. 4FF₁₆
(10. 89B)₁₆
(11. 8E9)₁₆
(12. C91)₁₆
(13. 109D)₁₆
(14. 1147)₁₆
(15. 25D9)₁₆
(16. 26D5)₁₆
(17. 2BCD)₁₆
(18. 4DE1)₁₆

(19. 54C5)₁₆
(1A. 5AA9)₁₆
(1B. ADD1)₁₆
(1C. 150E3)₁₆
(1D. 1AFA7)₁₆
(1E. 203D1)₁₆
(1F. 34C1B)₁₆
(20. B8C67)₁₆
(21. D1D29)₁₆
(22. 13313B)₁₆
(23. 1555FD)₁₆
(24. 2D69DD)₁₆

(25. 2E1A41)₁₆
(26. 6A64B1)₁₆
(27. CD7D25)₁₆
(28. 1405FAB)₁₆
(29. 16EC4E7)₁₆
(2A. 18C3197)₁₆
(2B. 1CFE799)₁₆
(2C. 1F12C01)₁₆
(2D. 236F73B)₁₆
(2E. 28AB159)₁₆
(2F. 291D8A1)₁₆
(30. 37341E9)₁₆

(31. 46C5031)₁₆
(32. 49A7B15)₁₆
(33. 4EC38ED)₁₆
(34. A068F8B)₁₆

[Dobri]

Originally Posted by tuckerkao

I can do the hex-base pretty well. The most memorable Hex Mersenne Prime is ADD1₁₆.

ADD1₁₆ = 44497₁₀

One can use Wolfram language (BaseForm[2^44497, 16]) to show the entire result (2000000000000...000000000000) in the base-16 numeral system. The first digit in this case is 2 followed by zeros.

[Dobri]

Originally Posted by WilliamJohnCox

My question is: are all known Mersenne Primes produced by Perfect Logarithmic Numbers, and if so, do they all begin with a 2 or N in UN, or 2 or 8 in ASCII?

Note that this is valid for ALL odd prime exponents but not just the Mersenne prime exponents.
For all odd primes p = 4k + 1, the first digit of 2^p in the base-16 numeral system is 2.
For all odd primes p = 4k + 3, the first digit of 2^p in the base-16 numeral system is 8.
The empirical observation here is that said first digit is more often 2 (p = 4k + 1) rather than 8 (p = 4k + 3) in the small sample of known Mersenne primes.

[Dr Sardonicus]

Originally Posted by WilliamJohnCox

In completing a paper for publication, Calculating the Geometry and Numbers of Universal Mathematics, which inserts the UNStar symbols for the powers of two in a alternative apt for ASCII: 1,2,3,U – 4,5,6,U – 7,8,9,S – C,X,W,10.

I just completed a table of the proportions of the Perfect Prime powers of two which produce the Perfect Logarithmic Numbers,
<snip>

I am unfamiliar with this terminology. Google Translate wasn't much help:

'Twas brillig, and the slithy toves did gyre and gimbal in the wabe...

[kriesel]

Originally Posted by Dr Sardonicus

I am unfamiliar with this terminology. Google Translate wasn't much help

( * F33 )

[Dobri]

Originally Posted by WilliamJohnCox

My question is: are all known Mersenne Primes produced by Perfect Logarithmic Numbers, and if so, do they all begin with a 2 or N in UN, or 2 or 8 in ASCII?

This actually is valid for ALL odd positive integers (both non-prime and prime).

For all odd integers p = 4k + 1, the first digit of 2^p in the base-16 numeral system is 2.
For instance, 9 = 3x3 = 4x2+1 (non-prime), and 2^9 = 512 = 200₁₆.

For all odd integers p = 4k + 3, the first digit of 2^p in the base-16 numeral system is 8.
For instance, 15 = 3x5 = 4x3+3 (non-prime), and 2^15 = 32768 = 8000₁₆.

[Viliam Furik]

Originally Posted by Dobri

This actually is valid for ALL odd positive integers (both non-prime and prime).

For all odd integers p = 4k + 1, the first digit of 2^p in the base-16 numeral system is 2.
For instance, 9 = 3x3 = 4x2+1 (non-prime), and 2^9 = 512 = 200₁₆.

For all odd integers p = 4k + 3, the first digit of 2^p in the base-16 numeral system is 8.
For instance, 15 = 3x5 = 4x3+3 (non-prime), and 2^15 = 32768 = 8000₁₆.

Yep, exactly. Because all Mersenne primes must have an odd number of ones (except M2 = 3) in their binary representation, specifically it must be either 4k+1 or 4k+3. Thus Mp + 1 has one 1, and p zeros. Every 4 consecutive bits make a hex digit, thus there are either two or four bits left in the front, making 2 (10) or 8 (1000) in hex.