Sentence: Every integer can be expressed as the sum of three perfect cubes.
My attempt: ∀x∈ℤ
and then I don't know what to do for "sum of three perfect cubes" part.
Sentence: Every integer can be expressed as the sum of two distinct integers
My attempt: ∀x∈ℤ
and then I don't know how to do the "sum of two distinct integers"
Sentence: There exists a largest integer.
My attempt: ∃x
and then I don't know how to "exists a largest integer"
$$\forall x \in \mathbb{Z}^+, \exists \{a,b,c\} \subset \mathbb{Z}\ {\rm s.t.}\ a^3 + b^3 + c^3 = x$$
$$\forall x \in \mathbb{Z}^+, \exists \{a, b \} \subset \mathbb{Z}\ {\rm s.t.}\ a \neq b \wedge (a + b = x)$$
$$\exists y \in \mathbb{Z}^+ {\rm s.t.}\ \forall x \neq y \in \mathbb{Z}^+ y > x$$
where of course $\mathbb{Z}$ represents the set of integers, and $\mathbb{Z}^+$ represents the set of positive integers, and ${\rm s.t.}$ means "such that."
One could use $(a,b,c) \in \mathbb{Z}^3$, rather than $\{ a,b,c \}$, but I prefer the curly brackets which are clearer usage for a set rather than a tuple, which might imply some structure. It means that each element in the set is in $\mathbb{Z}^+$.
Either way...