Statements in Peano arithmetic
• Sqrt is irrational:
Not[Exists[a, Exists[b, b!=0 && a × a == ΔΔ0 × (b × b)]]]
• There are infinitely many primes of the form n^2 + 1:
Not[Exists[n, ForAll[c, Exists[a, Exists[b, (n + c) × (n + c) + Δ0 == ΔΔa × ΔΔb]]]]]
• Every even number (greater than 2) is the sum of two primes (Goldbach's Conjecture; see page 135):
ForAll[a, Exists[b,Exists[c, ΔΔ0 × ΔΔa == b + c && ForAll[d,ForAll[e,ForAll[f,(f==ΔΔd × ΔΔe || f == Δ0) \[Implies] (f != b && f != 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 n^2+1 is prime.