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factorial and exponent

P: n/a
I want to calculate the value of 126 raise to the power 126 in turbo
C.
I've checked it with unsigned long int but it doesn't help.
So how could one calculate the value of such big numbers?
What's the technique?

Jun 16 '07 #1
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23 Replies


P: n/a

"Thomas" <my********************@gmail.comha scritto nel messaggio
news:11**********************@n60g2000hse.googlegr oups.com...
>I want to calculate the value of 126 raise to the power 126 in turbo
C.
I've checked it with unsigned long int but it doesn't help.
So how could one calculate the value of such big numbers?
What's the technique?
1. Use another programming language, or
2. find a bignum library, or
3. don't compute it. Compute its base-10 log. The integer part will
be the exponent, and from the fractional part you can find out the
mantissa.

<otlog10(126**126) = 126 * log10(126) </ot>
printf("%fe%d", pow(10, x - floor(x)), (int)floor(x));

where x is 126 * log10(126).

HTH.
Jun 16 '07 #2

P: n/a
In this article, I use ^ to represent "to the power of", rather than as
XOR.

Thomas said:
I want to calculate the value of 126 raise to the power 126 in turbo
C.
44329076602207821491972574571700100562486647339617 150064334557177890\
43517106373872170818953941792055669609014893218047 089803712563472169\
06583373889953014265747680923405829337012685381706 863104615274196776\
39132400195465417937691907225941135755503122280004 52759781376
I've checked it with unsigned long int but it doesn't help.
Since the largest value you are likely to be able to store in an
unsigned long int in Turbo C is 4294967295, it's hardly surprising that
you can't represent 126^126 in that type.
So how could one calculate the value of such big numbers?
What's the technique?
How would you do it by hand?

To save you some work, you'd probably start off by observing that
126^126 =
(126^63)^2 =
((126^31)^2*126)^2 =
(((126^15)^2*126)^2*126)^2 =
((((126^7)^2*126)^2*126)^2*126)^2 =
(((((126^3)^2*126)^2*126)^2*126)^2*126)^2 =
((((((126^2)*126)^2*126)^2*126)^2*126)^2*126)^2

So if you can multiply a number by itself, and multiply a number by 126,
you can get your result quite quickly.

See Knuth's "The Art of Computer Programming", volume 2, for information
on how to multiply two arbitrarily large numbers.

Alternatively, learn how to use GNU's GMP package, or Miracl, both of
which have C bindings.

--
Richard Heathfield
"Usenet is a strange place" - dmr 29/7/1999
http://www.cpax.org.uk
email: rjh at the above domain, - www.
Jun 16 '07 #3

P: n/a
Thomas wrote:
>
I want to calculate the value of 126 raise to the power 126 in
turbo C. I've checked it with unsigned long int but it doesn't
help. So how could one calculate the value of such big numbers?
What's the technique?
First, decide what holds the answer. You will need in the order of
1000 bits. Probably at least two of them.

--
<http://www.cs.auckland.ac.nz/~pgut001/pubs/vista_cost.txt>
<http://www.securityfocus.com/columnists/423>
<http://www.aaxnet.com/editor/edit043.html>
cbfalconer at maineline dot net
--
Posted via a free Usenet account from http://www.teranews.com

Jun 16 '07 #4

P: n/a
On Jun 16, 5:02 pm, Thomas <mynameisthomasander...@gmail.comwrote:
I want to calculate the value of 126 raise to the power 126 in turbo
C.
I've checked it with unsigned long int but it doesn't help.
So how could one calculate the value of such big numbers?
What's the technique?
Use GMP library found in http://gmplib.org/
It will enable you to do "Arithmetic without Limitations" !!

Jun 17 '07 #5

P: n/a
BiGYaN said:
On Jun 16, 5:02 pm, Thomas <mynameisthomasander...@gmail.comwrote:
>I want to calculate the value of 126 raise to the power 126 in turbo
C.
I've checked it with unsigned long int but it doesn't help.
So how could one calculate the value of such big numbers?
What's the technique?

Use GMP library found in http://gmplib.org/
It will enable you to do "Arithmetic without Limitations" !!
Nonsense.

Consider an integer greater than or equal to 2. Call it A. Consider
another integer greater than or equal to 2. Call it B.

Raise A to the power B, storing the result in A. Now raise B to the
power A, storing the result in B. If you repeat this often enough, you
*will* hit a limit, no matter what numerical library you use.

--
Richard Heathfield
"Usenet is a strange place" - dmr 29/7/1999
http://www.cpax.org.uk
email: rjh at the above domain, - www.
Jun 17 '07 #6

P: n/a

"Richard Heathfield" <rj*@see.sig.invalidha scritto nel messaggio
news:Gv******************************@bt.com...
BiGYaN said:
>On Jun 16, 5:02 pm, Thomas <mynameisthomasander...@gmail.comwrote:
>>I want to calculate the value of 126 raise to the power 126 in turbo
C.
I've checked it with unsigned long int but it doesn't help.
So how could one calculate the value of such big numbers?
What's the technique?

Use GMP library found in http://gmplib.org/
It will enable you to do "Arithmetic without Limitations" !!

Nonsense.

Consider an integer greater than or equal to 2. Call it A. Consider
another integer greater than or equal to 2. Call it B.

Raise A to the power B, storing the result in A. Now raise B to the
power A, storing the result in B. If you repeat this often enough, you
*will* hit a limit, no matter what numerical library you use.
But it is a limit of your computer, not of the library itself.
Jun 17 '07 #7

P: n/a
Army1987 wrote, On 17/06/07 09:48:
"Richard Heathfield" <rj*@see.sig.invalidha scritto nel messaggio
news:Gv******************************@bt.com...
>BiGYaN said:
>>On Jun 16, 5:02 pm, Thomas <mynameisthomasander...@gmail.comwrote:
I want to calculate the value of 126 raise to the power 126 in turbo
C.
I've checked it with unsigned long int but it doesn't help.
So how could one calculate the value of such big numbers?
What's the technique?
Use GMP library found in http://gmplib.org/
It will enable you to do "Arithmetic without Limitations" !!
Nonsense.

Consider an integer greater than or equal to 2. Call it A. Consider
another integer greater than or equal to 2. Call it B.

Raise A to the power B, storing the result in A. Now raise B to the
power A, storing the result in B. If you repeat this often enough, you
*will* hit a limit, no matter what numerical library you use.

But it is a limit of your computer, not of the library itself.
If it uses space allocated with malloc/realloc, then the library (rather
than the computer) has a limit because even with an infinite computer
size_t and pointers are of defined finite size, so you can only have a
block of known finite size and you can only chain a finite number of
such blocks together with pointers.

Of course, this applies to all libraries written in C.

It is also very important for people learning to be programmers (or who
already are programmers) to understand that in the real world resources
are always limited, so there is no such thing as "without limitations".
--
Flash Gordon
Jun 17 '07 #8

P: n/a
Army1987 said:
"Richard Heathfield" <rj*@see.sig.invalidha scritto nel messaggio
news:Gv******************************@bt.com...
>BiGYaN said:
<snip>
>>>
Use GMP library found in http://gmplib.org/
It will enable you to do "Arithmetic without Limitations" !!

Nonsense.

Consider an integer greater than or equal to 2. Call it A. Consider
another integer greater than or equal to 2. Call it B.

Raise A to the power B, storing the result in A. Now raise B to the
power A, storing the result in B. If you repeat this often enough,
you *will* hit a limit, no matter what numerical library you use.

But it is a limit of your computer, not of the library itself.
Nevertheless, it is a limit, and therefore the library *cannot* 'enable
you to do "Arithmetic without Limitations"', and therefore BiGYaN's
statement is nonsense.

Incidentally, you've just emerged from a 30-day spell in my sin bin. I
hope I won't have to chuck you back in there.

--
Richard Heathfield
"Usenet is a strange place" - dmr 29/7/1999
http://www.cpax.org.uk
email: rjh at the above domain, - www.
Jun 17 '07 #9

P: n/a
On Jun 17, 12:50 pm, Richard Heathfield <r...@see.sig.invalidwrote:
BiGYaN said:
On Jun 16, 5:02 pm, Thomas <mynameisthomasander...@gmail.comwrote:
I want to calculate the value of 126 raise to the power 126 in turbo
C.
I've checked it with unsigned long int but it doesn't help.
So how could one calculate the value of such big numbers?
What's the technique?
Use GMP library found inhttp://gmplib.org/
It will enable you to do "Arithmetic without Limitations" !!

Nonsense.

Consider an integer greater than or equal to 2. Call it A. Consider
another integer greater than or equal to 2. Call it B.

Raise A to the power B, storing the result in A. Now raise B to the
power A, storing the result in B. If you repeat this often enough, you
*will* hit a limit, no matter what numerical library you use.
"Arithmetic without Limitations" is sort of a slogan for GMP (http://
gmplib.org/). That's why I just put it in quotes.

The case that you are talking about does not show the limitation of
the numerical library. It's a limit of your computer. Besides, for all
*practical purposes* you won't hit this limit in a modern computer.
Like I'm quite sure that nobody will actually need all the digits of
126^126 for any *practical* job.

Jun 17 '07 #10

P: n/a
Richard Heathfield wrote:
Army1987 said:
>"Richard Heathfield" <rj*@see.sig.invalidha scritto nel messaggio
news:Gv******************************@bt.com...
>>BiGYaN said:
<snip>
>>>Use GMP library found in http://gmplib.org/
It will enable you to do "Arithmetic without Limitations" !!
Nonsense.

Consider an integer greater than or equal to 2. Call it A. Consider
another integer greater than or equal to 2. Call it B.

Raise A to the power B, storing the result in A. Now raise B to the
power A, storing the result in B. If you repeat this often enough,
you *will* hit a limit, no matter what numerical library you use.
But it is a limit of your computer, not of the library itself.

Nevertheless, it is a limit, and therefore the library *cannot* 'enable
you to do "Arithmetic without Limitations"', and therefore BiGYaN's
statement is nonsense.

Incidentally, you've just emerged from a 30-day spell in my sin bin. I
hope I won't have to chuck you back in there.
Of course there are limits, but I don't agree that they necessarily have
to be in the library. size_t is one limit, but if run on for example a
windows box it will not be *the* limit. A win32 application is not
allowed to allocate more than 2Gbytes of memory (and that's typically
half of what size_t allows for), unless you buy a more expensive version
of windows where that limit is raised to 3Gbytes.
It would also be possible for the mathematics library to internally use
something else than a standard C pointer and internally use paging
towards the system's hard disk or some internet based server or whatever
(magnetic tape?) allowing for a *much* higher limit. Oh well the limit
will still be there somewhere, but the calculation time will probably be
the limiting factor instead...

No, I don't seriously suggest using magnetic tape as a paging media...
but it would be possible!
Jun 17 '07 #11

P: n/a
Richard Heathfield wrote:
Army1987 said:
>"Richard Heathfield" <rj*@see.sig.invalidha scritto nel messaggio
news:Gv******************************@bt.com...
>>BiGYaN said:
<snip>
>>>Use GMP library found in http://gmplib.org/
It will enable you to do "Arithmetic without Limitations" !!
Nonsense.

Consider an integer greater than or equal to 2. Call it A. Consider
another integer greater than or equal to 2. Call it B.

Raise A to the power B, storing the result in A. Now raise B to the
power A, storing the result in B. If you repeat this often enough,
you *will* hit a limit, no matter what numerical library you use.
But it is a limit of your computer, not of the library itself.

Nevertheless, it is a limit, and therefore the library *cannot* 'enable
you to do "Arithmetic without Limitations"', and therefore BiGYaN's
statement is nonsense.
Of course there are limits, but I don't agree that they necessarily have
to be in the library. size_t is one limit, but if run on for example a
windows box it will not be *the* limit. A win32 application is not
allowed to allocate more than 2Gbytes of memory (and that's typically
half of what size_t allows for), unless you buy a more expensive version
of windows where that limit is raised to 3Gbytes.
It would also be possible for the mathematics library to internally use
something else than a standard C pointer and internally use paging
towards the system's hard disk or some internet based server or whatever
(magnetic tape?) allowing for a *much* higher limit. Oh well the limit
will still be there somewhere, but the calculation time will probably be
the limiting factor instead...

No, I don't seriously suggest using magnetic tape as a paging media...
but it would be possible!
Jun 17 '07 #12

P: n/a
BiGYaN said:
On Jun 17, 12:50 pm, Richard Heathfield <r...@see.sig.invalidwrote:
>BiGYaN said:
<snip>
Use GMP library found inhttp://gmplib.org/
It will enable you to do "Arithmetic without Limitations" !!

Nonsense.

Consider an integer greater than or equal to 2. Call it A. Consider
another integer greater than or equal to 2. Call it B.

Raise A to the power B, storing the result in A. Now raise B to the
power A, storing the result in B. If you repeat this often enough,
you *will* hit a limit, no matter what numerical library you use.

"Arithmetic without Limitations" is sort of a slogan for GMP (http://
gmplib.org/). That's why I just put it in quotes.
It's still false, within quotes or without them.
The case that you are talking about does not show the limitation of
the numerical library. It's a limit of your computer.
It's still a limit.
Besides, for all
*practical purposes* you won't hit this limit in a modern computer.
It's still a limit.
Like I'm quite sure that nobody will actually need all the digits of
126^126 for any *practical* job.
Cryptography springs to mind as a practical application which requires
exactness to the very last digit for calculations involving numbers of
that size and indeed greater.

--
Richard Heathfield
"Usenet is a strange place" - dmr 29/7/1999
http://www.cpax.org.uk
email: rjh at the above domain, - www.
Jun 18 '07 #13

P: n/a
Johan Bengtsson said:

<snip>
Of course there are limits, but I don't agree that they necessarily
have to be in the library.
I'm not saying they are, but that's not the issue. The claim was that
the library allows you to do arithmetic without limitations, and all
I'm saying is that that claim is false.

--
Richard Heathfield
"Usenet is a strange place" - dmr 29/7/1999
http://www.cpax.org.uk
email: rjh at the above domain, - www.
Jun 18 '07 #14

P: n/a
On Jun 18, 8:44 am, Richard Heathfield <r...@see.sig.invalidwrote:
Cryptography springs to mind as a practical application which requires
exactness to the very last digit for calculations involving numbers of
that size and indeed greater.
Thanks for informing .... I really had no idea. I take back my comment.

Jun 18 '07 #15

P: n/a

"Richard Heathfield" <rj*@see.sig.invalidha scritto nel messaggio news:A5******************************@bt.com...
Army1987 said:
>"Richard Heathfield" <rj*@see.sig.invalidha scritto nel messaggio
news:Gv******************************@bt.com...
>>BiGYaN said:
<snip>
>>>>
Use GMP library found in http://gmplib.org/
It will enable you to do "Arithmetic without Limitations" !!

Nonsense.

Consider an integer greater than or equal to 2. Call it A. Consider
another integer greater than or equal to 2. Call it B.

Raise A to the power B, storing the result in A. Now raise B to the
power A, storing the result in B. If you repeat this often enough,
you *will* hit a limit, no matter what numerical library you use.

But it is a limit of your computer, not of the library itself.

Nevertheless, it is a limit, and therefore the library *cannot* 'enable
you to do "Arithmetic without Limitations"', and therefore BiGYaN's
statement is nonsense.
If you cannot compute a number n with a computer, you can always
(at least in principle) use a computer with a larger size_t and
compute it.
Your statement is much like "You cannot use the long division
algorithm indefinitely because sooner or later you'll run out of
paper", or "There is a N such as you cannot draw a regular
(2^N * 3 * 5 * 17 * 257 * 65537)-gon with straightedge and compass,
because even if the polygon were as large as the universe, each
side would need to be shorter than a Planck length".
The library does enable Arithmetic without Limitations. It is the
implementation (and the universe) which put the limits.
Jun 24 '07 #16

P: n/a
JT
On Jun 24, 11:17 am, "Army1987" <please....@for.itwrote:
If you cannot compute a number n with a computer, you can always
(at least in principle) use a computer with a larger size_t and
compute it.
Your statement is much like "You cannot use the long division
algorithm indefinitely because sooner or later you'll run out of
paper", or "There is a N such as you cannot draw a regular
(2^N * 3 * 5 * 17 * 257 * 65537)-gon with straightedge and compass,
because even if the polygon were as large as the universe, each
side would need to be shorter than a Planck length".
The library does enable Arithmetic without Limitations. It is the
implementation (and the universe) which put the limits.
No. By your same argument, I can say this method
below "enables arithmetic without limitations":

int add(int a, int b) { return a+b; }
int sub(int a, int b) { return a-b; }

Because you can always build a C compiler that
provides a larger "int" size.

(For example, 32-bit C compilers use multiple
operations to simulate 64-bit integer operations.
The C compiler can double that up to simulate
128-bit, 256-bit, or in did even a much larger
bitwidth)

My two objections:

(1) That library does not "enable" unlimited arithmetic.
The library itself does not "impose" additional limit.

(2) People are confused between infinite,
and finite bounded. People should read more math books.

- JT
Jun 24 '07 #17

P: n/a
JT
On Jun 24, 11:46 am, JT <jackt...@gmail.comwrote:
(2) People are confused between infinite,
and finite bounded.
Sorry, of course, I meant "finite unbounded".

- JT
Jun 24 '07 #18

P: n/a

"JT" <ja******@gmail.comha scritto nel messaggio news:11*********************@q69g2000hsb.googlegro ups.com...
On Jun 24, 11:17 am, "Army1987" <please....@for.itwrote:
>If you cannot compute a number n with a computer, you can always
(at least in principle) use a computer with a larger size_t and
compute it.
Your statement is much like "You cannot use the long division
algorithm indefinitely because sooner or later you'll run out of
paper", or "There is a N such as you cannot draw a regular
(2^N * 3 * 5 * 17 * 257 * 65537)-gon with straightedge and compass,
because even if the polygon were as large as the universe, each
side would need to be shorter than a Planck length".
>The library does enable Arithmetic without Limitations. It is the
implementation (and the universe) which put the limits.
[snip]
My two objections:

(1) That library does not "enable" unlimited arithmetic.
The library itself does not "impose" additional limit.

(2) People are confused between infinite,
and finite unbounded. People should read more math books.
[correction incorporated above]

Indeed, I'm not saying that "Arithmetic without Limitations" means
that the library allows arithmetic with transfinite cardinals, only
that it allows arithmetic with arbitrarily large natural (finite)
numbers.
If there are indeed limits, they are due to the implementation.
Wait for a computer with more memory, and you'll be able to compute
larger numbers.

By your argument, the long division algorithm does not "enable" you
to divide arbitrarily large numbers, it just doesn't "impose"
additional limit (to that dictated by the size of the paper sheet
you work on).
Jun 24 '07 #19

P: n/a
Army1987 said:
"Richard Heathfield" ha scritto...
>Army1987 said:
>>"Richard Heathfield" ha scritto...
<snip>
>>>Raise A to the power B, storing the result in A. Now raise B to the
power A, storing the result in B. If you repeat this often enough,
you *will* hit a limit, no matter what numerical library you use.

But it is a limit of your computer, not of the library itself.

Nevertheless, it is a limit, and therefore the library *cannot*
'enable you to do "Arithmetic without Limitations"', and therefore
BiGYaN's statement is nonsense.
If you cannot compute a number n with a computer, you can always
(at least in principle) use a computer with a larger size_t and
compute it.
No, in principle you'll run out of resources at some point.
Your statement is much like "You cannot use the long division
algorithm indefinitely because sooner or later you'll run out of
paper",
Correct.
or "There is a N such as you cannot draw a regular
(2^N * 3 * 5 * 17 * 257 * 65537)-gon with straightedge and compass,
because even if the polygon were as large as the universe, each
side would need to be shorter than a Planck length".
Correct.
The library does enable Arithmetic without Limitations.
No, it doesn't. To do so, it would have to remove all limitations on
arithmetic, and it simply can't.
It is the
implementation (and the universe) which put the limits.
And therefore the limits are there. If the library does not remove them,
it does not enable arithmetic without limits.

--
Richard Heathfield <http://www.cpax.org.uk>
Email: -www. +rjh@
Google users: <http://www.cpax.org.uk/prg/writings/googly.php>
"Usenet is a strange place" - dmr 29 July 1999
Jun 24 '07 #20

P: n/a

"Richard Heathfield" <rj*@see.sig.invalidha scritto nel messaggio news:m9******************************@bt.com...
Army1987 said:
>or "There is a N such as you cannot draw a regular
(2^N * 3 * 5 * 17 * 257 * 65537)-gon with straightedge and compass,
because even if the polygon were as large as the universe, each
side would need to be shorter than a Planck length".
Correct.
So references which claim that a regular polygon of n sides is
constructible if and only if all the odd prime factors of n are
distinct Fermat primes (e.g. Wikipedia) must be wrong, since
2^100000000 * 3 * 17 * 257 is such a number, but such a polygon
cannot be constructed. :-)

(Or the limits of an algorithm are not the same thing as the limits
of its implementation, nor even the same thing as the limits of the
universe.)
Jun 24 '07 #21

P: n/a
Army1987 said:

<snip>
So references which claim that a regular polygon of n sides is
constructible if and only if all the odd prime factors of n are
distinct Fermat primes (e.g. Wikipedia) must be wrong, since
2^100000000 * 3 * 17 * 257 is such a number, but such a polygon
cannot be constructed. :-)
That depends on their definition of "constructible". As for Wikipedia
being wrong, that wouldn't particularly shock me.

--
Richard Heathfield <http://www.cpax.org.uk>
Email: -www. +rjh@
Google users: <http://www.cpax.org.uk/prg/writings/googly.php>
"Usenet is a strange place" - dmr 29 July 1999
Jun 24 '07 #22

P: n/a
On Jun 16, 7:02 am, Thomas <mynameisthomasander...@gmail.comwrote:
I want to calculate the value of 126 raise to the power 126 in turbo
C.
I've checked it with unsigned long int but it doesn't help.
So how could one calculate the value of such big numbers?
What's the technique?
Hi. This is similar to a programming project I'm doing in assembler.

I think the first thing you need to do, and I think someone else
mentioned this, is to find out the size of the final result. Then
make sure you feed the result there. You do this by using natural
logarithms, but I forget how, I had to ask my son. Convert 126^126
base ten = 2^ whatever.

I think you might consider bit shifting since 126 = 128 - 2.

128 = 1000 0000. So that would be shift left seven.
2 = 10. Shift left once.

Initialize your source=126. What you do is shift your source left
seven, add to a scratch area, shift it right 6, subtract from scratch,
voila you've just multiplied your source by 126. This becomes your new
source. Loop 126 times.
Jun 27 '07 #23

P: n/a
Tom Gear wrote On 06/27/07 14:21,:
On Jun 16, 7:02 am, Thomas <mynameisthomasander...@gmail.comwrote:
>>I want to calculate the value of 126 raise to the power 126 in turbo
C.
I've checked it with unsigned long int but it doesn't help.
So how could one calculate the value of such big numbers?
What's the technique?


Hi. This is similar to a programming project I'm doing in assembler.

I think the first thing you need to do, and I think someone else
mentioned this, is to find out the size of the final result. Then
make sure you feed the result there. You do this by using natural
logarithms, but I forget how, I had to ask my son. Convert 126^126
base ten = 2^ whatever.
You should be able to do this in your head, to a
reasonable approximation.

lg(126^126)
= 126*lg(126)
~= 126*lg(128)
= 126 * 7
= 882

Replacing 126 by 128 errs on the high side, so the
approximation cannot be too small. (It turns out --
I cheated and used a calculator -- that 880 bits will
suffice; the estimate is high by <0.23%.)
I think you might consider bit shifting since 126 = 128 - 2.
[...]
See TAOCP section 4.6.3 for efficient computation of
powers.

--
Er*********@sun.com
Jun 27 '07 #24

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