## Arithmetic puzzle

A forum for good logic/math puzzles.

Moderators: jestingrabbit, Moderators General, Prelates

houlahop
Posts: 30
Joined: Thu Jul 13, 2017 9:57 am UTC

### Arithmetic puzzle

22!=1124000727777607680000
The number 22! is a 22 digit number

11240
00727
77760
76800
00

Find a number n such as n! is n^2 digit number

More generally find n such as n! is n^k digit number (k>2)

jaap
Posts: 2091
Joined: Fri Jul 06, 2007 7:06 am UTC
Contact:

### Re: Arithmetic puzzle

It is not possible.

For n>1 we have:
log n! < log nn = n log n < n2

So n! always has fewer than n2 digits (except for 1! and 0!).

houlahop
Posts: 30
Joined: Thu Jul 13, 2017 9:57 am UTC

### Re: Arithmetic puzzle

Thank you

(n^2)/d(n!) is equal to pi(n) when n goes to infinity (where pi(n) is the counting function of primes)

Is there any interpretation of this "equality"?

My last post because I wanted to point out to this.
Good luck and good bye!
To the moderator :
Conclusion : Either you did not read the post, either you are lying.
Where did you see any contempt from myself?

SecondTalon
SexyTalon
Posts: 26428
Joined: Sat May 05, 2007 2:10 pm UTC
Location: Louisville, Kentucky, USA, Mars. HA!
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### Re: Arithmetic puzzle

Alright then. Bye.
heuristically_alone wrote:I want to write a DnD campaign and play it by myself and DM it myself.
heuristically_alone wrote:I have been informed that this is called writing a book.

Schrödinger's Wolves
Posts: 14
Joined: Fri Mar 24, 2017 3:28 am UTC

### Re: Arithmetic puzzle

houlahop wrote:Thank you

(n^2)/d(n!) is equal to pi(n) when n goes to infinity (where pi(n) is the counting function of primes)

Is there any interpretation of this "equality"?

My last post because I wanted to point out to this.
Good luck and good bye!
To the moderator :
Conclusion : Either you did not read the post, either you are lying.
Where did you see any contempt from myself?

I would guess that interpretation is lim(n->infinity) (n^2)/(d(n!)*pi(n))=1.
What is the function d in this case?

Xias
Posts: 363
Joined: Mon Jul 23, 2007 3:08 am UTC
Location: California
Contact:

### Re: Arithmetic puzzle

I believe d(n) is just shorthand for "the number of digits in n."

I wonder about the fact that n^2 and pi(n) are not dependent on the base, but d(n) is.