Beautiful factorial equation

Number Theory Level 3

$\large(1+x!)(1+y!)=(x+y)!$

How many non-negative integral ordered pairs $(x,y)$ satisfy the equation above?

This is a part of my set "Beautiful.. It is!" .

The answer is 2.

1 solution

Ravi Dwivedi
Jul 10, 2015

Notice that for $x,y \geq 2, (1+x!)$ and $(1+y!)$ are both odd but $(x+y)!$ is even. So the equation has no solutions for $x,y \geq 2$

When $x=1$ the equation is

$2(1+y!)=(1+y)!$

Notice that $y=2$ is a solution.If $y \geq 3$ then 3 divides RHS but not LHS

So $y \geq 3$ is not a solution when $x=1$

and $y=1$ is not a solution in this case( $x=1)$

Hence $x=1,y=2$ and $x=2,y=1$ due to symmetry of the equation.

So there are $2$ non negative integral ordered pairs

Moderator note:

Good case checking. Note that "the equation has no solutions for $x, y, \geq 2$ " does not imply that " $(x,y) = (1,1)$ is the remaining case to check".

For x=1 we have checked for all non negative y and by symmetry y=1 is also checked

Ravi Dwivedi - 5 years, 11 months ago

Beautiful factorial equation

This is a part of my set "Beautiful.. It is!" .

The answer is 2.

1 solution

Moderator note:

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