20210806, 13:07  #1 
Aug 2021
2_{10} Posts 
Are all MP# produced by logarithmic numbers beginning with 2 or 8 in ASCII?
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 30page Introduction to Universal Quantum Numbers can be accessed at https://williamjohncox.com/UNIntro.pdf. Thanks, ~wm 
20210806, 14:53  #2  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{2}·13·127 Posts 
Quote:
Plus, the ambiguous repeated symbol "U" makes it all look very silly. 

20210806, 16:33  #3  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
3^{2}×761 Posts 
Quote:
From the linked pdf: "decimal base10 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 selfpublishing component. Last fiddled with by kriesel on 20210806 at 16:45 

20210807, 05:05  #4  
"Καλός"
May 2018
5·73 Posts 
Quote:
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 base16 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 base16 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 

20210807, 06:21  #5 
"Tucker Kao"
Jan 2020
Head Base M168202123
2×389 Posts 
Hi Dobri, I can do the hexbase pretty well. The most memorable Hex Mersenne Prime is ADD1_{16}.
1. 2 2. 3 3. 5 4. 7 5. D 6. 11_{16} 7. 13_{16} 8. 1F_{16} 9. 3D_{16} A. 59_{16} B. 6B_{16} C. 7F_{16} D. 209_{16} E. 25F_{16} F. 4FF_{16} (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} 
20210807, 06:45  #6  
"Καλός"
May 2018
5·73 Posts 
Quote:
One can use Wolfram language (BaseForm[2^44497, 16]) to show the entire result (2000000000000...000000000000) in the base16 numeral system. The first digit in this case is 2 followed by zeros. 

20210807, 13:34  #7  
"Καλός"
May 2018
5·73 Posts 
Quote:
For all odd primes p = 4k + 1, the first digit of 2^p in the base16 numeral system is 2. For all odd primes p = 4k + 3, the first digit of 2^p in the base16 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. 

20210807, 14:02  #8  
Feb 2017
Nowhere
5992_{10} Posts 
Quote:
'Twas brillig, and the slithy toves did gyre and gimbal in the wabe... 

20210807, 14:44  #9 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
3^{2}·761 Posts 

20210807, 15:54  #10  
"Καλός"
May 2018
5×73 Posts 
Quote:
For all odd integers p = 4k + 1, the first digit of 2^p in the base16 numeral system is 2. For instance, 9 = 3x3 = 4x2+1 (nonprime), and 2^9 = 512 = 200_{16}. For all odd integers p = 4k + 3, the first digit of 2^p in the base16 numeral system is 8. For instance, 15 = 3x5 = 4x3+3 (nonprime), and 2^15 = 32768 = 8000_{16}. 

20210807, 16:48  #11  
"Viliam Furík"
Jul 2018
Martin, Slovakia
19·41 Posts 
Quote:


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