Curious and want to share about Prime number 23
 

Hi,
While running a program for a diffferent reason, I found that prime number 23 has a peculiar character.

Statement: if p is a PRIME number and n is the Number of digits for Factorial(p), then n = p will be true for 23 Only.

Result:
P = 23
Fact(23) = 25,852,016,738,884,976,640,000 (23 digits)

I tried upto Fact(31667), and haven't found anything.

If this is something known or proven earleir, pls accept my apologies.
Thanks a lot!

It is indeed the only such prime, (see [URL="http://www.research.att.com/%7Enjas/sequences/table?a=34886&fmt=4"]http://www.research.att.com/~njas/sequences/table?a=34886&fmt=4[/URL], a(n) means the number of decimal digits in n!; n=a(n) only for n=22, 23, and 24, and 23 is the only prime of the 3) but it's not really an interesting fact. There's no reason to only consider a prime for such a property, and noting that the only times where n=a(n) are 22, 23, and 24 is not terribly interesting.

You are not the first to observe this. [URL="http://primes.utm.edu/curios/page.php/23.html"]http://primes.utm.edu/curios/page.php/23.html[/URL] says: "23 is the only prime number p such that p! is p digits long. [Gupta]"

n! = n*(n-1)! so if n>=10 then n! has at least one more digit than (n-1)!
25! has 26 digits so n! must always have more digits than n for n>=25.

A more interesting (albeit trivial) generalization is to find solution(s) using aribitrary base, with a prime or with any n. Find the smallest base in which there's no solution. Find a smallest base with exactly two solutions (or prove that there isn't such). etc etc etc

The number 23 has [url=http://en.wikipedia.org/wiki/The_Number_23]a [b]lot[/b] of properties[/url].