How to write these statements in a predicate logic form

Question

1. There is one and only one natural number $n$ for which the proposition formula $P(n)$ holds.

2. There are no integer solutions to the equations $x^2-y^2=11$

Answer 1

(I think that you meant predicate logic, not propositional logic.)

For (1) the trick is to do it in two parts: first specify that there’s at least one $n$ that works, and then specify that it’s the only $n$ that works. Thus, you want to start with the proposition $$\exists n\big(n\in\mathbb{N}\land P(n)\big)\;.$$ To say that there’s only one such $n$, you need to add a clause saying that every $m$ satisfying $P$ is equal to the specific one whose existence you just asserted: $$\exists n\bigg(n\in\mathbb{N}\land P(n)\land\forall m\Big(\big(m\in\mathbb{N}\land P(m)\big)\to m=n\Big)\bigg)\;.$$ In words this boils down to:

There is a natural number $n$ satisfying $P$ such that if $m$ is any natural number satisfying $P$, then $m=n$.

For (2), consider how you’d express the statement that there is at least one integer solution to the equation $x^2-y^2=11$; that would be

$$\exists n \exists m(n\in\mathbb{N}\land m\in\mathbb{N}\land\dots)\;,\tag{1}$$

where I leave you to fill in the blank. The statement that you actually want says that $(1)$ is false; how do you modify $(1)$ to say that?

(By the way, is (2) true, or false?)