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-   -   2^64 is not a big number! (https://www.mersenneforum.org/showthread.php?t=26854)

 Bobby Jacobs 2021-05-30 18:17

2^64 is not a big number!

We have searched all primes up to 2[SUP]64[/SUP]=18446744073709551616, but that is not a big number. If something had 2[SUP]64[/SUP] atoms in it, then it would be like a small grain of sand. 2[SUP]64[/SUP] is a small number when it comes to atoms.

 xilman 2021-05-30 19:32

[QUOTE=Bobby Jacobs;579502]We have searched all primes up to 2[SUP]64[/SUP]=18446744073709551616, but that is not a big number. If something had 2[SUP]64[/SUP] atoms in it, then it would be like a small grain of sand. 2[SUP]64[/SUP] is a small number when it comes to atoms.[/QUOTE]About a milligram of sand if I calculate correctly.

Easily visible but not especially large --- about 1mm across.

 rudy235 2021-05-30 19:52

Bobby: There is very little correlation between [I]small numbers[/I] in the Physical world and "[I]small numbers[/I]" when it comes to searching for primes.

PRIMO has the capability of proving primes in the range of 10[SUP]40000[/SUP] to 10[SUP]50000[/SUP]

Lucas-Lehmer can prove Mersenne primes in the order of 2[SUP]100,000,000[/SUP] to probably 2[SUP]1000,000,000[/SUP] (with current technolog).

In the physical world [B]Atoms in the planet Earth[/B]. (approx) 10[SUP]52[/SUP]

[B]Atoms in the Solar System[/B] (if it is conceived as a solid sphere of 50 Light Year Radius is of the order of 10[SUP]69[/SUP]

Even if you go the [B]atoms in all know Undiverse[/B] which is estimated to have Radius of 4.65*10[SUP]10[/SUP] light-years, the number of atoms in that volume would "only be" 2*10[SUP]106[/SUP]

So, in conclusion even a number as "[I]small[/I]" as 10[SUP]120[/SUP] would have no equivalent in the physical world.

You can try creating sort of fancy artificial numbers: for instance the number of distinct molecules theoretically possible by combining up to 1000 atoms of Carbon, Oxygen, Nitrogen, Hydrogen, Chlorine, Magnesium, Iron, Flour, Calcium and Sodium atoms but even that won't get you any closer the the smallest composite number that has not been factored which is [URL="https://en.wikipedia.org/wiki/RSA_numbers#RSA-260"][FONT="Arial Black"][SIZE="3"]RSA-260[/SIZE][/FONT][/URL]

(Of course they are millions of smaller numbers that have not been factored, but that is because no serious effort has been applied to them.)

 charybdis 2021-05-30 20:08

[QUOTE=Bobby Jacobs;579502]We have searched all primes up to 2[SUP]64[/SUP]=18446744073709551616, but that is not a big number.[/QUOTE]

Depends who you're talking to...

[quote=Daniel Schroeder, An Introduction to Thermal Physics]
[B]Small numbers[/B] are small numbers, like 6, 23, and 42. You already know how to manipulate small numbers.
[B]Large numbers[/B] are much larger than small numbers, and are frequently made by exponentiating small numbers...
The most important property of large numbers is that you can add a small number to a large number without changing it. For example,
[$]10^{23}+23 = 10^{23}[/$]
...
[B]Very large numbers[/B] are even larger than large numbers, and can be made by exponentiating large numbers. An example would be [$]10^{10^{23}}[/$]. Very large numbers have the amazing property that you can multiply them by large numbers without changing them. For instance,
[$]10^{10^{23}} \times 10^{23} = 10^{10^{23}+23} = 10^{10^{23}}[/$]
[/quote]

 sweety439 2021-05-30 23:08

[QUOTE=Bobby Jacobs;579502]We have searched all primes up to 2[SUP]64[/SUP]=18446744073709551616, but that is not a big number. If something had 2[SUP]64[/SUP] atoms in it, then it would be like a small grain of sand. 2[SUP]64[/SUP] is a small number when it comes to atoms.[/QUOTE]

People have found (but not stored) all the primes up to 2^64, this is because small primes are too easy to find. They can be found far faster than they can be read from a hard disk, however, this is a [URL="https://github.com/xayahrainie4793/small-primes/archive/refs/heads/main.zip"]zipped list for all the primes up to 358*2^25[/URL]

 retina 2021-05-31 00:25

I've seen estimates of the total number of configurations (permutations or orderings?) of all particles in the observable universe is ~10^360.

What about TREE(3)?
[youtube]3P6DWAwwViU[/youtube]

Or TREE(G)?
[youtube]0X9DYRLmTNY[/youtube]

Or TREE(G)^^...^^TREE(G)?

What do you compare it to? Compared to infinity, all numbers you can think of will be insignificant and lost in the rounding error.

 sweety439 2021-05-31 02:23

[QUOTE=retina;579525]Or TREE(G)^^...^^TREE(G)?

What do you compare it to? Compared to infinity, all numbers you can think of will be insignificant and lost in the rounding error.[/QUOTE]

This number is less than TREE(TREE(TREE(...TREE(TREE(TREE(G)))...))) with G TREE's

 a1call 2021-05-31 04:02

If you started drawing 10 short lines/notches every second 24/7, it would take you well over 7 billion years to finish drawing 2^64 lines.:smile:
Light travels 0.3 Micrometers (1/1000 of a millimeter) in 1 femtosecond.
Light will travel more than 18446 Light-Years in 2^64 femtoseconds. This is more than 4000 times the distance to the closest stars to our sun.

ETA OTOH, If you could fold a piece of paper (in half) a mere 64 times it would have 2^64 layers. The Samurai-Swords (as well as the Chinese-Noodles) are folded about 50 times and stretched/flattened each time. This gives the sword an edge which is about one molecule thick.:smile:

 gd_barnes 2021-06-01 04:18

[QUOTE=a1call;579535]If you started drawing 10 short lines/notches every second 24/7, it would take you well over 7 billion years to finish drawing 2^64 lines.:smile:[/QUOTE]

Yes it's definitely over 7 billion years but...it's much worse than that:

2^64 = 18,446,744,073,709,551,616 (~1.8447 * 10^19)

2^64 / 3600 seconds per hour / 24 hour per day / ~365.25 days per year / = ~584,542,046,090.6 years.

At 10 lines / second it would be 1/10th that length but would still be ~58,454,204,609.06 or ~58.454 billion years!

Since 10 lines per second seems a little faster than the average person can write...I would go with 1 line per second, which would take ~584.542 billion years!!

Either way it's likely longer than the universe has been around.

:smile:

 Xyzzy 2021-06-01 11:47

[QUOTE]Everywhere is within walking distance if you have the time.[/QUOTE] - Steven Wright

 xilman 2021-06-01 12:57

I'd like to see you walk from the Earth to the Moon.

 Viliam Furik 2021-06-01 14:05

[QUOTE=xilman;579653]I'd like to see you walk from the Earth to the Moon.[/QUOTE]

Note the usage of words - "walking [B]distance[/B]".

 LaurV 2021-06-01 14:17

[QUOTE=xilman;579653]I'd like to see you walk from the Earth to the Moon.[/QUOTE]
Give me the time...

 Bobby Jacobs 2021-06-03 17:23

[QUOTE=a1call;579535]
Light travels 0.3 Micrometers (1/1000 of a millimeter) in 1 femtosecond.
Light will travel more than 18446 Light-Years in 2^64 femtoseconds. This is more than 4000 times the distance to the closest stars to our sun.
[/QUOTE]

Actually, light travels about 18446 light-seconds in 2[SUP]64[/SUP] femtoseconds. That is less than a light-year.

 a1call 2021-06-03 20:59

Thank you very much for the correction :smile:

[CODE]2^64*0.3 micrometers = 5534023222112865484.8 micrometers = 5534023222112.8654848 m = 5534023222.1128654848 km = 0.00058493005201488906931613994292358101681 ly
[/CODE]

 a1call 2021-06-03 22:55

[QUOTE]
The "back half of the chessboard" is a reference to the old story about the inventor of chess. As the story goes, when chess was presented to a great king, the king offered the inventor any reward that he wanted. The inventor asked that a single grain of rice be placed on the first square of the chessboard. Then two grains on the second square, four grains on the third, and so on. Doubling each time.

The king, baffled by such a small price for a wonderful game, immediately agreed, and ordered the treasurer to pay the agreed upon sum. A week later, the inventor went before the king and asked why he had not received his reward. The king, outraged that the treasurer had disobeyed him, immediately summoned him and demanded to know why the inventor had not been paid. The treasurer explained that the sum could not be paid - by the time you got even halfway through the chessboard, the amount of grain required was more than the entire kingdom possessed.

The king took in this information and thought for a while. Then he did the only rational thing a king could do in those circumstances. He had the inventor killed, as an object lesson in the perils of trying to outwit the king.
[/QUOTE]

[url]https://www.forbes.com/sites/alexknapp/2011/11/17/the-seduction-of-the-exponential-curve/[/url]

 robert44444uk 2021-06-05 06:54

[QUOTE=LaurV;579667]Give me the time...[/QUOTE]

I feel confident that there are people who do the distance in a lifetime. But probably not most specimens of homo sapiens species.

People are encouraged to do 10,000 steps a day in the UK, and an average step is about 70 cm. Allowing for pre toddler days and bedridden days at the far end, a good mobile life (say the top 10%-15% of the population) is 85 years in G20 countries.

So a first approximation:

85*365.25*10000*0.7/1000 = 217,300 km in a lifetime

The moon is Km 356,500 closest distance from the earth centre to centre, or approx. 348,400 km surface to surface.

There will be fit pastoralists in Africa who achieve this, I am sure. Not so sure about us car-obsessed people in the G20. Some will though.

 Bobby Jacobs 2021-06-07 19:49

I recall the story of the grains of rice. The first square on the chessboard had 1 grain of rice, the second square had 2 grains, the third square had 4 grains, ..., the last square had 2[SUP]63[/SUP]=9223372036854775808 grains. In all, there were 2[SUP]64[/SUP]-1=18446744073709551615 grains of rice. That might seem like a lot of rice, but there are more atoms in a single grain of rice than the number of grains of rice in that story.

 chalsall 2021-06-07 20:05

[QUOTE=Bobby Jacobs;580264]...but there are more atoms in a single grain of rice than the number of grains of rice in that story.[/QUOTE]

There are more possible positions in the game of Go than there are atoms in the Universe.

That's why [URL="https://www.youtube.com/watch?v=WXuK6gekU1Y"]AlphaGo[/URL] was so impressive. And scary (for those who like humans).

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