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How do I write Fermat's Last Theorem using only universal and existential quantifiers.

I only want to quantify it over the natural numbers (excluding zero)- for which I use $ℤ+$ to represent this set. My attempt at it, which I don't think is right, is this:

$∀(x∈ℤ+) ∀(y∈ℤ+) ∀(z∈ℤ+) ∃(n∈ℤ+) (x^n+y^n=z^n → n<3)$

R.McGinnis
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  • can't you negate the existential? I would go with $$\forall x,y,z\in\mathbb{Z}+,\quad \nexists n\in\mathbb{Z}{\geq 3},\quad x^n+y^n=z^n.$$ – Integral Nov 10 '16 at 01:01
  • I think the $\exists n$ should be $\forall n$, like: $∀(x∈ℤ+) ∀(y∈ℤ+) ∀(z∈ℤ+) ∀(n∈ℤ+) (x^n+y^n=z^n → n<3)$. It says, whatever $x,y,z,n$ are, IF $x^n+y^n=z^n$ THEN $n$ must be at most $2$. – Mirko Nov 10 '16 at 01:36
  • Yes, what Mirko says! – Bram28 Nov 10 '16 at 01:45
  • Thanks all for your answers. It's been 20+ years since I've done this and the memory of it is a little clouded- but from your answers and thinking more about it I've got it. – R.McGinnis Nov 11 '16 at 22:22

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