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tony123

Member

Registered: 2007-08-03

Posts: 229

Find all positive integers

Find all positive integers

and

which satisfy

.

Last edited by tony123 (2008-08-20 09:38:04)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Find all positive integers

Since all factorials are positive, x must be strictly greater than the other four letters.
(Otherwise one of the letters would be greater than or equal to x, and so you'd get x! = x! + k or x! = (x+n)! + k, neither of which are possible)

So then the maximum of all the RHS variables is x-1, meaning the RHS's maximum value is 4(x-1)!.

Since x! = x(x-1)!, this means that x≤4.

x is now between 1 and 4. The other variables are all between 1 and x, which means there are a finite amount of combinations to try. It's now fairly easy to see that the only solutions are:

x = 3; {f,g,y,z} = {1,1,2,2}
x = 4; {f,g,y,z} = {3,3,3,3}

There are 6 ways that the first way can happen, which gives 7 solutions in all.
If the positive restriction on the integers was dropped, the set {1,1,2,2} could have one of both of its 1's replaced with zero(es), which would give two more solution sets and 18 more solutions.

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It wanted to be normal.