Statements in Peano arithmetic
Examples include:
• √2\!\(\*SqrtBox[\(2\)]\) is irrational:
¬ ∃a (∃b (b ≠ 0 ∧ a × a (Δ Δ 0) × (b × b)))\[Not]\!\(\*SubscriptBox[\(\[Exists]\),\(a\)]\)\!\(\*SubscriptBox[\(\[Exists]\),\(b\)]\)(b\[NotEqual]0\[And]a\[Times]a==(\[CapitalDelta]\[CapitalDelta]0)\[Times](b\[Times]b)))
• There are infinitely many primes of the form n2 + 1\!\(\*SuperscriptBox[\(n\),\(2\)]\)+1:
¬ ∃n (∀c (∃a (∃b (n + c) × (n + c) + Δ 0 (Δ Δ a) × (Δ Δ b))))\[Not]\!\(\*SubscriptBox[\(\[Exists]\),\(n\)]\)(\!\(\*SubscriptBox[\(\[ForAll]\),\(c\)]\)(\!\(\*SubscriptBox[\(\[Exists]\),\(a\)]\)(\!\(\*SubscriptBox[\(\[Exists]\),\(b\)]\)(n+c)\[Times](n+c)+\[CapitalDelta]0==(\[Delta]\[Delta])\[Times](\[CapitalDelta]\[CapitalDelta]b))))
• Every even number (greater than 2) is the sum of two primes (Goldbach's Conjecture; see page 135):
∀a (∃b (∃c((Δ Δ 0) × (Δ Δ a) b + c ∧ ∀d (∀e (∀f ((f (Δ Δ d) × (Δ Δ e) ∨ f Δ 0) ⇒ (f ≠ b ∧ f ≠ c)))))))\!\(\*SubscriptBox[\(\[ForAll]\),\(a\)]\)(\!\(\*SubscriptBox[\(\[Exists]\),\(b\)]\)(\!\(\*SubscriptBox[\(\[Exists]\),\(c\)]\)((\[CapitalDelta]\[CapitalDelta]0)\[Times](\[CapitalDelta]\[CapitalDelta]a)==b+c\[And]\!\(\*SubscriptBox[\(\[ForAll]\),\(d\)]\)(\!\(\*SubscriptBox[\(\[ForAll]\),\(e\)]\)(\!\(\*SubscriptBox[\(\[ForAll]\),\(f\)]\)((f==(\[CapitalDelta]\[CapitalDelta]d)\[Times](\[CapitalDelta]\[CapitalDelta]e)\[Or]==\[CapitalDelta]0)\[Implies](f\[NotEqual]b\[And]f\[NotEqual]c)))))))
The last two statements have never been proved true or false, and remain unsolved problems of number theory. The picture shows spacings between n for which n2 + 1\!\(\*SuperscriptBox[\(n\),\(2\)]\)+1 is prime.