" only prime number that can only be divided by itself and one" logic

Question

How do I write "only prime number that can only be divided by itself and one" and I can only use these predicates : H(x, y) is x/y=integer(x mod y=0), S(x, y) is x=y, and P(x) is x is prime number? My guess is : ∀x∀y(~S(x, 1)^(~S(x, y)^~S(y,1)→~H(x, y))→P(x)) But I can't check if it's true or false

Answer 1

The statement: "only prime number that can only be divided by itself and one" is not well formed.   It's a bit garbled.   I suspect you meant: "the only numbers that can divide a prime number are one and itself," or perhaps "a prime number can only be divided by itself and one."

Which means that you want to say: "for any integer $x$ and any integer $y$, if $x$ is a prime integer and $y$ is neither $1$ nor $x$, then $y$ will not divide $x$".

Can you?

$$\forall x\,\forall y\,\Big(\big(P(x)\wedge \neg S(y,x)\wedge\neg S(y,1)\big)\to \neg H(x,y)\Big)\\\forall x\forall y \Big(\big(P(x)\wedge H(x,y)\big)\to \big(S(y,x)\vee S(y,1)\big)\Big)$$