https://en.wikipedia.org/wiki/Largest_known_prime_number wrote:
<<As of January 2018, the largest known prime number is 2^{77,232,917} − 1, a number with 23,249,425 digits. It was found in December 2017 by the Great Internet Mersenne Prime Search (GIMPS). A standard word processor layout (50 lines per page, 75 digits per line) would require 6,199 pages to display it. Its value is:
467333183359231099988335585561115521251321102817714495798582338593567923480521177207484311099740208849621368090038049317... (23,249,185 digits omitted) ...285376004518786055402223376672925679282131965467343395945397370476369279894627999939614659217371136582730618069762179071 >>
50th Mersenne Prime found!

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50th Mersenne Prime found!
Art Neuendorffer

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Re: 50th Mersenne Prime found!
Base 10 is just too insufficient for such enormous numbers. I looked into higher bases once when Art had made a similar post, but wasn't able to find anything higher than hexadecimal (base 16) in common use. What's called for is a much higher base system, say, base 100, in which there would be 100 characters to represent the base ten numbers 0 to 99. Such a system would be able to compress the exact expression of a humongously long real number into a more manageable size.
Have any large base systems been developed?
Bruce
Have any large base systems been developed?
Bruce
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Re: 50th Mersenne Prime found!
It would allow the number to be printed using less paper. I don't see what other value it would have. Would you or I have a better grasp of the number if it were notated in base 100?BDanielMayfield wrote:What's called for is a much higher base system, say, base 100, in which there would be 100 characters to represent the base ten numbers 0 to 99. Such a system would be able to compress the exact expression of a humongously long real number into a more manageable size.
No, going the other way is what makes the most sense. Expressing it in binary. Because once a number becomes to large to make any intuitive sense expressed in base 10, it's probably only being manipulated in a computer, anyway, and for that, base 2 is the natural system.
(The ancient Babylonians utilized base 60, sexagesimal. AFAIK that's the largest base ever used for common math.)
Chris
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Re: 50th Mersenne Prime found!
Being able to accurately express something in a much more efficient manner has value. Using less paper is good.Chris Peterson wrote:It would allow the number to be printed using less paper. I don't see what other value it would have. Would you or I have a better grasp of the number if it were notated in base 100?BDanielMayfield wrote:What's called for is a much higher base system, say, base 100, in which there would be 100 characters to represent the base ten numbers 0 to 99. Such a system would be able to compress the exact expression of a humongously long real number into a more manageable size.
15_{(base 10)} = F_{(base 16)} = 1111_{(base 2)}, therefore hex is more efficient than the decimal and especially the binary systems, at least in character count. The 50th Mersenne Prime would require a staggering number of digits in binary.No, going the other way is what makes the most sense. Expressing it in binary. Because once a number becomes to large to make any intuitive sense expressed in base 10, it's probably only being manipulated in a computer, anyway, and for that, base 2 is the natural system.
Very good to know. Thanks.(The ancient Babylonians utilized base 60, sexagesimal. AFAIK that's the largest base ever used for common math.)
Bruce
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Re: 50th Mersenne Prime found!
15(base 10) = F(base 60) too.
There are lots of base conversion tools available online  if you really have nothing else to do.
Rob
There are lots of base conversion tools available online  if you really have nothing else to do.
Rob

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Re: 50th Mersenne Prime found!
And yet, that's precisely how it was found. Binary computation. Internal binary representation. (Computers don't use hexadecimal.)BDanielMayfield wrote:15_{(base 10)} = F_{(base 16)} = 1111_{(base 2)}, therefore hex is more efficient than the decimal and especially the binary systems, at least in character count. The 50th Mersenne Prime would require a staggering number of digits in binary.
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Re: 50th Mersenne Prime found!
BDanielMayfield wrote:
The 50th Mersenne Prime would require a staggering number of digits in binary.
 77,232,917 bits to be precise:
but, at least, it would be easy to remember (so long as you were allowed to count on your fingers).
Art Neuendorffer

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Re: 50th Mersenne Prime found!
Bit. As in binary digit.neufer wrote:BDanielMayfield wrote: The 50th Mersenne Prime would require a staggering number of digits in binary.
 77,232,917 bits to be precise:
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51th Mersenne Prime found!
https://en.wikipedia.org/wiki/Largest_known_prime_number wrote:
<<The largest known prime number (as of January 2020) is 2^{82,589,933} − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. Its value is:
148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560... (24,861,808 digits omitted)
...062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591.>>
Art Neuendorffer

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Re: 51th Mersenne Prime found!
That's a funny shape of that curve. Reminds me somehow of the Universe with expands faster or slower during different epochs.neufer wrote: ↑Fri Jul 31, 2020 6:26 pmhttps://en.wikipedia.org/wiki/Largest_known_prime_number wrote:
<<The largest known prime number (as of January 2020) is 2^{82,589,933} − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. Its value is:
148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560... (24,861,808 digits omitted)
...062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591.>>
Although, unlike the Universe, the prime number curve seems to be slowing down.
Ann
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Re: 51th Mersenne Prime found!
Careful... the yaxis is logarithmic. The declining slope of the curve only indicates "slowing down" in certain contexts.Ann wrote: ↑Fri Jul 31, 2020 7:42 pmThat's a funny shape of that curve. Reminds me somehow of the Universe with expands faster or slower during different epochs.neufer wrote: ↑Fri Jul 31, 2020 6:26 pmhttps://en.wikipedia.org/wiki/Largest_known_prime_number wrote:
<<The largest known prime number (as of January 2020) is 2^{82,589,933} − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. Its value is:
148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560... (24,861,808 digits omitted)
...062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591.>>
Although, unlike the Universe, the prime number curve seems to be slowing down.
Ann
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Flattening the Curve
Right Chris. It would become a favorite tool of flat Earth type thinkers, if they could grasp the concept. Most any curve can be flattened simply by switching to a log or semilog graph.Chris Peterson wrote: ↑Fri Jul 31, 2020 7:54 pmCareful... the yaxis is logarithmic. The declining slope of the curve only indicates "slowing down" in certain contexts.
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Re: 51th Mersenne Prime found!
The Universe had 2 kicks in the butt after the Big Bang:
 Inflation.
Dark Energy.
 Electronic computers ~1960
Massively parallel computers ~1985
Large scale distributed computing projects ~2000
Perhaps (some subset of) the Mersenne prime exponents have some (as yet) undetermined regularity:
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, ...
(Blue color: proven consecutive Mersenne prime exponents.)
Art Neuendorffer