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 2015-04-01, 23:29 #1 henryzz Just call me Henry     "David" Sep 2007 Cambridge (GMT/BST) 585310 Posts Factorial puzzle I just noticed that 10!=6!*7!. Are there any other instances where x!=y!*z! with y and z > 1? If not can anyone prove this?
 2015-04-01, 23:41 #2 henryzz Just call me Henry     "David" Sep 2007 Cambridge (GMT/BST) 585310 Posts Just realised that there is an infinite number of examples where (n!)!=n!*(n!-1)! Are there any more apart from this trivial form?
2015-04-02, 00:22   #3
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

20C016 Posts

Quote:
 Originally Posted by henryzz Just realised that there is an infinite number of examples where (n!)!=n!*(n!-1)! Are there any more apart from this trivial form?
well one condition on this occuring is that one of the factorials involved has to involve the last prime below the factorial being divided by other factorials so for example the condition for example the reason 10!=6!*7! works is because one of the factorials involves 7 a prime number no solution can leave this out. also a key part is that the other factorial has to include possible factorizations of the rest so 8*9*10 = 8*90 = (2*4)*(5*6*3) which means it can barely fall below sqrt(number) when trying to find solutions to number! okay I made an error but I was trying to limit the cases.

Last fiddled with by science_man_88 on 2015-04-02 at 00:53

 2015-04-02, 02:27 #4 science_man_88     "Forget I exist" Jul 2009 Dumbassville 26×131 Posts
2015-04-02, 12:02   #5
ATH
Einyen

Dec 2003
Denmark

37·83 Posts

Quote:
 Originally Posted by henryzz I just noticed that 10!=6!*7!. Are there any other instances where x!=y!*z! with y and z > 1? If not can anyone prove this?
No other examples for y,z < 10000 except the trivial cases.

2015-04-02, 12:58   #6
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts

Quote:
 Originally Posted by ATH No other examples for y,z < 10000 except the trivial cases.
using pari I've checked for solutions using up to the last 10000 factorials before a factorial up to 30000! I think:

Code:
factorfactorial(e=0)={
a=parvector(10000,n,(n+e)!);
parfor(x=10,#a,
d=precprime(x);x-d,
c,
if(c==0||c>=10000,next());
b=setintersect(vector(c-1,t,a[x]/a[x+t-(c+1)]),a);
if(b==[],,
print(x","b)
)
);
e=e+#a;
print(e);
factorfactorial(e)
};
doh I see an error now and it might of been slowing it down.

Last fiddled with by science_man_88 on 2015-04-02 at 13:58

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