My Number is Bigger! (Everymen Needed!)

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username5243
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My Number is Bigger! (Everymen Needed!)

Postby username5243 » Sat Dec 19, 2015 11:09 am UTC

Not to be confused with PsiSquared's game. This one requires more average joe participation.

Recently there was some discussion in the other thread about how the everymen left at around the same place. This thread attempts to fix that. I've shamelessly copypasted PsiSquared's rules from the other game.

The rules:
1. Post a number which is bigger than the previous number.
2. Each entry must define a single computable positive number.
3. You're not allowed to reference previously posted numbers directly (you can't post stuff like "the sum of all previous numbers plus one" or "double the previous number").
4. You can use any notation you want, but it must be understandable by the average math-oriented joe who has no specific background in "Big Number-ology". I know that many large-numbers enthusiasts love to create (or copy) incredibly obscure notations without explaining them properly, but this is not the place for it. Here, you should strive to be as clear as possible (Note to the experts: This does not - in any way - limit the size of the numbers you can reach. It simply means that as the game progresses, we need to be clearer with our explanations).
5. If you're "a math-oriented average joe" and an entry seems completely unintelligible to you, you may challenge it. You may not continue to submit numbers until all challenges have been resolved.
6. People are encouraged to ask questions about the numbers and notations posted.
7. Borrowing notations from other players is allowed. In this case, the "borrower" does not need to explain the notation again.
8. Don't make too big a jump. The idea is to progress gradually, so even the uninitiated could feel at home.
9. If you need a general (and rough) guideline: A good number to reach at the end of page 1 would be a million. And a good number for the end of page 2 would be a googol. Note that this is only a guideline, and not a hard limit. The only reason I've posted it is so you'll have a better idea of what I mean by "too big a jump", which is - obviously - a subjective term.
10. Since there is no hard-set "Cap" for the numbers one may post, I reserve the right to disqualify blatant overkills. I will, however, do this only in the most extreme of situations (and sincerely hope such drastic measures will never be needed).
11. All posters are either "googologists" or "everymen". A player specifies which one they are once they make their first post, and may never switch. It is the responsibility of the googologists to explain things to the everymen.
12. At least one in every ten submitted numbers must be made by an everyman. That is, if the last nine numbers were posted by googologists, a fifth number may not be posted by a googologist.

And now, I'll start nice and easy:

I am a googologist.

...And send:

1
Last edited by username5243 on Wed Jan 13, 2016 1:04 am UTC, edited 2 times in total.
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Re: My Number is Bigger! (Everyman Version)

Postby emlightened » Sat Dec 19, 2015 1:20 pm UTC

You might want to advertise the differences in the thread title (maybe an [everymen needed!]) from the other thread, to show the differences. Also, you might want to clean up Rule 2, and merge 5/6 with 11/12/13.

(I am a 'googologist'.) Googology is the study of very large numbers. A googologist is someone who does googology a lot. For now, I'd say people would probably want to be in the latter category if they understand how 3↑↑↑↑3 works, and how to compare it to 2↑↑↑↑4, for instance. If you don't understand it... well we'll be teaching how it works in a few pages time, so you should soon.

Edit: I'm not sure if making the googologist/everyman stuff permanent is a good idea. Some people will have a great understanding of ↑ and → arrows, but not a clue when having to deal with arrays, or a bad understanding initially but then study into large-ordinal level stuff. I guess it's just worth saying that, in these cases, username might see to make exceptions to the rule.

π2
Last edited by emlightened on Sat Dec 19, 2015 1:31 pm UTC, edited 1 time in total.

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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Re: My Number is Bigger! (Everyman Version)

Postby username5243 » Sat Dec 19, 2015 1:30 pm UTC

2

Okay, thanks.
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Re: My Number is Bigger! (Everyman Version)

Postby PsiSquared » Sat Dec 19, 2015 8:58 pm UTC

emlightened wrote:I'm not sure if making the googologist/everyman stuff permanent is a good idea. Some people will have a great understanding of ↑ and → arrows, but not a clue when having to deal with arrays, or a bad understanding initially but then study into large-ordinal level stuff. I guess it's just worth saying that, in these cases, username might see to make exceptions to the rule.


It's might also come off as condescending to some people.

I would have thrown that distinction altogether, had the rule of "only 4 googologist entries in a row are allowed" not been so indispensable for this version of the game.

Anyway:

3
(That's four in row. So this is the last consecutive entry allowed by our lot)

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Sat Dec 19, 2015 9:39 pm UTC

Alright, now we've got to wait for someone else to post. There shouldn't be any complaints yet.
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Re: My Number is Bigger! (Everymen Needed!)

Postby thecamoninja » Sun Dec 20, 2015 9:45 am UTC

I'd like to volunteer myself as a hapless everyperson who's very much interested in/intrigued by math, but honestly not very good at it and certainly not too knowledgeable. And, now that I'm in college and no one can ever force me to take a math class again, what little I know is sure to slip away. I don't think I'm really familiar with any notations other than regular old scientific notation (although I'm sure I could obtain a decent understanding by googling up a youtube video or two).

That said, 101
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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Sun Dec 20, 2015 6:48 pm UTC

Hi, thecamoninja. I hope that we can stop that from happening for another few months. :P

a↑b = ab (useful for nesting exponentiation, and other stuff that we'll come to later)

5↑1.5 = 51.5 ~ 11.18

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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Re: My Number is Bigger! (Everymen Needed!)

Postby Vytron » Sun Dec 20, 2015 6:49 pm UTC

Hi Camo. So... I invented a notation and it was banned from another thread because it was claimed that it was too complicated for the everyman to understand. If over the course of this game you were to "get it" then I could make a case about it, or something.

It starts like this:

n, = (n),

Any number that has a comma afterwards gets enclosed by parenthesis.

,x = (x),x

If nothing appears before the comma, it means that the whole thing after the comma is its input.

(n), = ((n+1)↑(n+1))+1

Where ↑ is exponentiation or iterated multiplication, like this:

a↑b = a*a*a... for b "a"s

Your number would be expressed like this:

101 = 10↑1 = 10 (i.e. 10*10*10... for one 10)

Examples:

0, = (0), = ((0+1)↑(0+1))+1 = (1↑1)+1 = (1)+1 = 2
1, = (1), = ((1+1)↑(1+1))+1 = (2↑2)+1 = (2*2)+1 = (4)+1 = 5
2, = (2), = ((2+1)↑(2+1))+1 = (3↑3)+1 = (3*3*3)+1 = (27)+1 = 28
3, = (3), = ((3+1)↑(3+1))+1 = (4↑4)+1 = (4*4*4*4)+1 = (256)+1 = 257
4, = (4), = ((4+1)↑(4+1))+1 = (5↑5)+1 = (5*5*5*5*5)+1 = (3125)+1 = 3126

And so on.

Is the notation clear so far?

I'm a googologist.

Send:
2,
= 28
Last edited by Vytron on Sun Dec 20, 2015 6:56 pm UTC, edited 1 time in total.

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Re: My Number is Bigger! (Everymen Needed!)

Postby Vytron » Sun Dec 20, 2015 6:54 pm UTC

emlightened wrote:5↑1.5 = 51.5 ~ 11.18


Can you explain how does it work? I.e. how can you multiply 5 by itself 1.5 times?

I think this explanation would be useful to have definitions for all of these:

5↑1.5 = something
5↑↑1.5 = something
5↑↑↑1.5 = something
...

By using the same fundamental procedure that 5↑1.5 uses for multiplication.

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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Sun Dec 20, 2015 7:25 pm UTC

Making hyperoperators (+, ×, ↑, ↑↑ etc.) continuous has never been easy, although you can certainly approximate them easily. Exponentiation is the last one we've been able to do, and the calculation I did was 5↑1.5 = 5×(5↑0.5) = 5×(√5) ~ 5×2.236 = 11.18. The reason we can calculate ↑ continuously, but not other higher hyperoperators, is because we can calculate × continuously, and we also know that bc+d = bc×bd, which means that we can calculate exponentiation continuously by setting c=d and c+d=n to figure out square roots, or multiply results together to add the exponents, which will get us as close as we want to any real number. However, no similar law applies to larger hyperoperators (because a×b = b×a but a↑b ≠ b↑a and a↑↑b ≠ b↑↑a etc., generally) so we can't extend them this way.

However, we can approximate 5↑1.5 fairly easily: 5↑1.5 = 5×(5↑0.5) ~ 5×(1+(5-1)×0.5) = 5×3 = 15, although fairly inaccurate, this can be extended to tetration etc. (the weird 1+(5-1)×k was needed because it needs to rise linearly between 5↑1=5 and 5↑2=25; this can be simplified for tetration (↑↑) etc.)

2↑5 = 32

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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Sun Dec 20, 2015 8:45 pm UTC

The answer to life, the universe and everything:
42

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Sun Dec 20, 2015 8:55 pm UTC

And that's four.
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Re: My Number is Bigger! (Everymen Needed!)

Postby thecamoninja » Sun Dec 20, 2015 9:36 pm UTC

Vytron, your notation kind of makes sense to me? But it seems a little bit needlessly overcomplicated, at least for small numbers. I suppose it might work more clearly for very large ones? I mainly don't get why you add in all the +1s.

Also, is there any particular reason to use the upwards arrow rather than the carat (^)? I'm a fan of the carat myself, and I was under the impression it was the standard if you weren't using superscripts.

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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Sun Dec 20, 2015 9:56 pm UTC

thecamoninja wrote:Also, is there any particular reason to use the upwards arrow rather than the carat (^)? I'm a fan of the carat myself, and I was under the impression it was the standard if you weren't using superscripts.


It's because it's the standard that Knuth introduced for tetration, pentation etc., and a lot of us have gotten used to using it for exponents too, instead of the carat. It's not wrong to use either, it's just preference.

91.71
Last edited by emlightened on Sun Dec 20, 2015 10:04 pm UTC, edited 1 time in total.

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Sun Dec 20, 2015 10:02 pm UTC

It's standard, but the caret is often used in ASCII approximations. I think everyone will understand if you use that instead.

Anyway, I'll start a notation for this thread.

Define x>y as:

10^x for y = 0
10>(y-1) for x = 0
((x-1)>y)>(y-1) for both x and y not equal to 0

For now, all you need to know is that n>0 = 10^n. I just defined the rest of it to save time.

Anyway I send:

2>0

Which is equal to 10^2 = 100.
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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Sun Dec 20, 2015 10:18 pm UTC

And now for my notation, which is a simple extention of good-old scientific notation:

Ex = 10^x
aEn = a * 10^n

Note that for integer x, Username5243's " x > 0 " is equal to my Ex.

So:

E3 = 10^3 = 1000 (a thousand)
E6 = 10^6 = 1,000,000 (a million)
4E7 = 4*10^7 = 40,000,000

Everything clear so far?


If so, I send:

2E2
( = 200)


(Note to those who know my letter-notation from other threads: I will not use it in its entirety here)

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Sun Dec 20, 2015 10:23 pm UTC

While we're here we might as well get started with tetration.

Tetration is written as a↑↑b or a^^b. It means a^a^...^a^a^a with b a's. Tetration is always evaluated right to left.

So:

2↑↑3 = 2^2^2 = 2^4 = 16

3↑↑2 = 3^3 = 27

4↑↑2 = 4^4 = 256

And send:

4↑↑2 = 256
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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Sun Dec 20, 2015 10:24 pm UTC

And that's four.

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Re: My Number is Bigger! (Everymen Needed!)

Postby SirGabriel » Sun Dec 20, 2015 11:30 pm UTC

I'm an everyman.

17^2 = 289

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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Sun Dec 20, 2015 11:32 pm UTC

3E2 = 300

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Mon Dec 21, 2015 1:26 am UTC

400
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Re: My Number is Bigger! (Everymen Needed!)

Postby Yrgos » Tue Dec 22, 2015 6:50 pm UTC

I teach math at high-school level, but in terms of comparing big numbers I am a complete novice. Thus I am a math interessed everyman.

Vytron: most of your notation was clear, but this part " ,x = (x),x " does not leave me with any onsight as to how I would evaluate for example:
,3 = (3),3 = 4^4+1,3 = 257,3 but where would I go from there? I must say I like functions where it is clear what is the input and what is the output :)

Send:
5^4 = (5^2)*(5^2) = 25*25 = 625

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Tue Dec 22, 2015 8:08 pm UTC

Welcome!

Send:

666
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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Tue Dec 22, 2015 8:09 pm UTC

Vytron hasn't yet posted the definition of x,3 :-)

His notation is an hierarchy of (yet to be defined) functions:

x,
x,0
x,0<stuff>
x,1
x,1<stuff>
and so on.

Up until now, he only defined the first of these functions: x, .

So we don't yet have the definitions required to calculate (3),3. Nor do we need them, because even 1,1 is supposed to be an insanely mindbogglingly huge number which won't be relevant for another dozen pages.

Anyway:
E3 = 1000

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Tue Dec 22, 2015 8:12 pm UTC

And I was going to use 3>0 = 10^3...

Anyway...

E(pi) = 10^pi ~ 1385.455731367...

I wanted to use E(e) (which is about 522.7353), but that's too small.
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Re: My Number is Bigger! (Everymen Needed!)

Postby Yrgos » Tue Dec 22, 2015 9:28 pm UTC

I have some ideas for a function, but it'll have to wait till I have a computer I can use. I support the use of scientific notation, that is familiar territory :)

Send:
2^11 = 2048 or 2,048E3

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Re: My Number is Bigger! (Everymen Needed!)

Postby SirGabriel » Tue Dec 22, 2015 9:36 pm UTC

12345

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Re: My Number is Bigger! (Everymen Needed!)

Postby Yrgos » Tue Dec 22, 2015 9:42 pm UTC

Send:
Number of 5 card hands you can create from useing only the red cards in a standard 52 card deck.
That is 26 choose 5. Or 26c5 = 26!/(5!(26-5)!) = 26×25×24×23×22/(5×4×3×2) = 65780

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Tue Dec 22, 2015 9:48 pm UTC

77777
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Re: My Number is Bigger! (Everymen Needed!)

Postby Yrgos » Tue Dec 22, 2015 9:54 pm UTC

Send:
The number of 5 card hands from 3 suits in a regular 52 card deck.
575 757
Pritty number :)

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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Tue Dec 22, 2015 11:24 pm UTC

For those that have been watching the other thread, I'm only going to use this one for a bit.

My notation works like this:

If you've got a number, and it's got a {0} to the left of it (but nothing to the right), you delete the {0} and add one to the number (so {0}3 = 4).
If you've got a number, and it's got a {1} to the left of it (but nothing to the right), you delete the {1} and replace it with that number of {0}'s (so {1}3 = {0}{0}{0}3).
If you've got a number, and it's got a {2} to the left of it (but nothing to the right), you delete the {2} and replace it with that number of {1}'s (so {2}3 = {1}{1}{1}3).
Generally, if you've got a number with a {n+1} to the left of it (but nothing to the right), you delete the {n+1} and replace it with that number of {n}'s (so {n+1}3 = {n}{n}{n}3).
To make it more concise, instead of writing out {1}{1}{1}, you can write out 3{1}, for instance. i.e. a number to the left of {n} represents that number of copies of {n} (so 6{1}6 = {1}{1}{1}{1}{1}{1}6).

For {0}, {1} and {2}, we can write their values explicitly, easily.
{0}n = n+1
{1}n = {0}{0}...{0}n (with n {0}'s) = 2n
{2}n = {1}{1}...{1}n (with n {1}'s) = n*2^n

{2}4{1}1 = {2}3{1}2 = {2}2{1}4 = {2}{1}8 = {2}16 = 16*2^16 = 1048576

Is anything unclear?

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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Re: My Number is Bigger! (Everymen Needed!)

Postby Yrgos » Tue Dec 22, 2015 11:55 pm UTC

Seems kinda clear, would:
{3}n then be n{2}n = n*2^(n^2) ?

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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Wed Dec 23, 2015 12:10 am UTC

No, it's quite a bit larger than that.

2{2}n = {2}(n*2^n) = (n*2^n)*2^(n*2^n) > 2^2^n
3{2}n = 2{2}(n*2^n) = {2}(n*2^n)*2^(n*2^n) = ((2nn)*2^(2nn))*2^((2nn)*2^(2nn)) > 2^2^2^n
{3}n = n{2}n > 2^2^...2^n (for n 2^'s) > 2↑↑n

For instance, {3}4 = 4{2}4 = 3{2}(4*2^4) = 3{2}64 = 2{2}(64*2^64) = 2{2}(2^70) = {2}((2^70)*2^(2^70)) ≈ {2}2^2^70 = ((2^2^70)*2^(2^2^70)) ≈ 2^2^2^70

The thing to look at is the n in the 2^n part of that result. If you replace that with n*2^n, as in 2{2}, the equation grows like a double exponential (2^(n*2^n) ≈ 2^2^n), and if you replace it again, it grows like a triple exponential, etc. When looking at the power of functions, it's often best to focus on the most powerful part of the function (the exponent, in this case), and iterate that; this simplified function is (almost) always as powerful as the original.

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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Re: My Number is Bigger! (Everymen Needed!)

Postby Yrgos » Wed Dec 23, 2015 12:49 am UTC

Ahh, I didn't substitute both n's! :D
Now that function grows insanely fast! oO
But it's approwed by an everyman. (y)

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Re: My Number is Bigger! (Everymen Needed!)

Postby Yrgos » Wed Dec 23, 2015 1:05 am UTC

To beat your number I'll bust out the whole deck!
Send: number of different 5 card hands from a deck of 52 different cards.
2 598 960

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Re: My Number is Bigger! (Everymen Needed!)

Postby SirGabriel » Wed Dec 23, 2015 2:08 am UTC

e^e^e ≈ 3814279.1

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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Wed Dec 23, 2015 2:17 am UTC

E6.6 = 106.6 ~ 3 981 071.

Note that unlike ordinary scientific notation, you're allowed to put non-integers after my "E". For now this is nothing more than a neat quirk, but it will be very useful when I define the next level of my notation.

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Wed Dec 23, 2015 11:51 am UTC

5000000
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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Wed Dec 23, 2015 2:50 pm UTC

{2}{0}{1}9 = {2}19 = 19*219 = 9,961,472

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Wed Dec 23, 2015 2:55 pm UTC

Send:

7>0

= 10^7 = 10,000,000

And that's four in a row.
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