50th Mersenne Prime found!

[neufer]

[b][color=#0000FF][size=125]Plot of the number of digits in largest known prime by year, since the electronic computer. The vertical scale is logarithmic.[/size][/color][/b]

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<<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 >>

[BDanielMayfield]

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

[Chris Peterson]

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.

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?

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.)

[BDanielMayfield]

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.

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?

Being able to accurately express something in a much more efficient manner has value. Using less paper is good.

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.

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.

(The ancient Babylonians utilized base 60, sexagesimal. AFAIK that's the largest base ever used for common math.)

Very good to know. Thanks.

Bruce

[rstevenson]

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

[Chris Peterson]

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.

And yet, that's precisely how it was found. Binary computation. Internal binary representation. (Computers don't use hexadecimal.)

[neufer]

The 50th Mersenne Prime would require a staggering number of digits in binary.

• 77,232,917 bits to be precise:

111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111... (77,232,677 bits omitted) ...111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

but, at least, it would be easy to remember (so long as you were allowed to count on your fingers).

[Chris Peterson]

The 50th Mersenne Prime would require a staggering number of digits in binary.

• 77,232,917 bits to be precise:

Bit. As in binary digit.

[neufer]

A 2016 plot of the number of digits in largest known prime by year, since the electronic computer. The vertical scale is logarithmic.

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<<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.>>

[Ann]

A 2016 plot of the number of digits in largest known prime by year, since the electronic computer. The vertical scale is logarithmic.

---

<<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.>>

That's a funny shape of that curve. Reminds me somehow of the Universe with expands faster or slower during different epochs.

Although, unlike the Universe, the prime number curve seems to be slowing down.

Ann

[Chris Peterson]

A 2016 plot of the number of digits in largest known prime by year, since the electronic computer. The vertical scale is logarithmic.

---

<<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.>>

That's a funny shape of that curve. Reminds me somehow of the Universe with expands faster or slower during different epochs.

Although, unlike the Universe, the prime number curve seems to be slowing down.

Ann

Careful... the y-axis is logarithmic. The declining slope of the curve only indicates "slowing down" in certain contexts.

[BDanielMayfield]

That's a funny shape of that curve. Reminds me somehow of the Universe with expands faster or slower during different epochs.

Although, unlike the Universe, the prime number curve seems to be slowing down.

Ann

Careful... the y-axis is logarithmic. The declining slope of the curve only indicates "slowing down" in certain contexts.

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 semi-log graph.

[neufer]

A 2016 plot of the number of digits in largest known prime by year, since the electronic computer. The vertical scale is logarithmic.

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That's a funny shape of that curve. Reminds me somehow of the Universe with expands faster or slower during different epochs. Although, unlike the Universe, the prime number curve seems to be slowing down.

The Universe had 2 kicks in the butt after the Big Bang:

• Inflation.
  Dark Energy.

Mersenne prime discovery has had 3 kicks in the butt in the last hundred years:

• Electronic computers ~1960
  Massively parallel computers ~1985
  Large scale distributed computing projects ~2000

It is unclear where the next kick in the butt will come from. Quantum computing

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.)