Why is a (factorial) rarely a square ?
In article <4973fd1a-3e40-4214-993d-96247bd175e8n@googlegroups.com>, henh...@gmail.com <henhanna@gmail.com> wrote:
Why is a (factorial) rarely a square ?
For n>1, n! is never a square. This follows trivially [walks away,
returns a couple of minutes later], yes trivially, from Bertrand's
postulate.
In article <4973fd1a-3e40-4214-993d-96247bd175e8n@googlegroups.com>, henh...@gmail.com <henhanna@gmail.com> wrote:
Why is a (factorial) rarely a square ?
For n>1, n! is never a square.
This follows trivially [walks away,
returns a couple of minutes later], yes trivially, from Bertrand's
postulate.
On 26/07/2022 8:13 pm, Richard Tobin wrote:
In article <4973fd1a-3e40-4214...@googlegroups.com>,
henh...@gmail.com <henh...@gmail.com> wrote:
Why is a (factorial) rarely a square ?
For n>1, n! is never a square.I note in passing that it comes close four or five times, notably
7!, which is just 1 shy of 71^2.
I further note in passing that the subject line asks about n!-1
rather than n! itself, but I tried it both ways and it turns out
to matter little.
This follows trivially [walks away,I think I can trivialise this still further.
returns a couple of minutes later], yes trivially, from Bertrand's postulate.
#include <stdio.h>
#include <limits.h>
int main(void)
{
printf("%lu\n", ULONG_MAX);
return 0;
}
yields 18446744073709551615, the largest available positive
integer without going to ridiculous (and indeed non-trivial)
extremes like departing from C89.
The largest non-trivial factorial is therefore 20! or
2432902008176640000.
(21 is 51090942171709440000, which is simply too big to be
trivial; let us not exceed our brief.)
1! = 1 IS a square: root 1^2 is out by 0
2! = 2 is NOT a square: root 1^2 is out by 1
3! = 6 is NOT a square: root 2^2 is out by 2
4! = 24 is NOT a square: root 5^2 is out by 1
5! = 120 is NOT a square: root 11^2 is out by 1
6! = 720 is NOT a square: root 27^2 is out by 9
7! = 5040 is NOT a square: root 71^2 is out by 1
8! = 40320 is NOT a square: root 201^2 is out by 81
9! = 362880 is NOT a square: root 602^2 is out by 476
10! = 3628800 is NOT a square: root 1905^2 is out by 225
11! = 39916800 is NOT a square: root 6318^2 is out by 324
I ran it all the way up to and including 20. After 11!, the
discrepancy between the factorial and the square of its nearest
integer root (which discrepancy must be 0 for a solution)
continues to climb by amounts that increase by, on average, close
to an order of magnitude per factorial.
And, unless you want to get non-trivial, them's all the numbers
there is. QED.
--
10! = 3628800 is NOT a square: root 1905^2 is out by 225
11! = 39916800 is NOT a square: root 6318^2 is out by 324
For n>1, n! is never a square. This follows trivially [walks away,
returns a couple of minutes later], yes trivially, from Bertrand's
postulate.
I have a truly marvelous proof of Mr Tobin's statement, which
Usenet is too narrow to contain.
In article <tbphev$25ufh$1@dont-email.me>,
Richard Heathfield <rjh@cpax.org.uk> wrote:
For n>1, n! is never a square. This follows trivially [walks away,
returns a couple of minutes later], yes trivially, from Bertrand's
postulate.
I have a truly marvelous proof of Mr Tobin's statement, which
Usenet is too narrow to contain.
The basic idea is simple: any prime factor of a square must occur
twice (or 4 times, or 6 ...).
Why is a (factorial) rarely a square ?
Why is a (factorial - 1) rarely a square ?
Why is a (factorial + 1) rarely a square ?
In article <4973fd1a-3e40-4214...@googlegroups.com>,
henh...@gmail.com <henh...@gmail.com> wrote:
Why is a (factorial) rarely a square ?For n>1, n! is never a square. This follows trivially [walks away,
returns a couple of minutes later], yes trivially, from Bertrand's
postulate.
-- Richard
Sysop: | Keyop |
---|---|
Location: | Huddersfield, West Yorkshire, UK |
Users: | 325 |
Nodes: | 16 (2 / 14) |
Uptime: | 64:26:51 |
Calls: | 7,124 |
Calls today: | 2 |
Files: | 12,524 |
Messages: | 5,521,314 |