Statements in Peano arithmetic

Examples include:

• √2 is irrational:

¬ ∃_{a} (∃_{b} (b ≠ 0 ∧ a × a (Δ Δ 0) × (b × b)))

• There are infinitely many primes of the form n^{2} + 1:

¬ ∃_{n} (∀_{c} (∃_{a} (∃_{b} (n + c) × (n + c) + Δ 0 (Δ Δ a) × (Δ Δ 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)))))))

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.