Finding an $a$, such that $\forall x(x^x-(a\cdot x)!=0).$

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My previous question was wrongly formulated, I wanted to know the value of $a$ as $x$ gets bigger, but due to my limited math knowledge I can't solve it. So, is $a$ a constant and what would its value be?

Thank you

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So, is a a constant ?

No. The factorial or $\Gamma$ function cannot be expressed in terms of elementary functions, which is what would happen if such an a were to exist.