Curious and want to share about Prime number 23

spkarra · 2009-11-20, 00:56

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!

Mini-Geek · 2009-11-20, 01:09

It is indeed the only such prime, (see http://www.research.att.com/~njas/sequences/table?a=34886&fmt=4, 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.

Jens K Andersen · 2009-11-20, 02:20

You are not the first to observe this. http://primes.utm.edu/curios/page.php/23.html 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.

Batalov · 2009-11-20, 03:39

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

ixfd64 · 2009-11-20, 03:54

The number 23 has a lot of properties.

2009-11-20, 00:56	#1
spkarra "Sastry Karra" Jul 2009 Bridgewater, NJ (USA) 3³ Posts	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!

2009-11-20, 02:20	#3
Jens K Andersen Feb 2006 Denmark 2×5×23 Posts	You are not the first to observe this. http://primes.utm.edu/curios/page.php/23.html 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. Last fiddled with by Jens K Andersen on 2009-11-20 at 02:21 Reason: fix typo*

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2009-11-20, 01:09	#2
Mini-Geek Account Deleted "Tim Sorbera" Aug 2006 San Antonio, TX USA 1000010101011₂ Posts	It is indeed the only such prime, (see http://www.research.att.com/~njas/sequences/table?a=34886&fmt=4, 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.

2009-11-20, 03:39	#4
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2·7·677 Posts	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

2009-11-20, 03:54	#5
ixfd64 Bemusing Prompter "Danny" Dec 2002 California 2³×13×23 Posts	The number 23 has a lot of properties.