Theorem 1: For all n > 0

∑_{i=1}^{n} i = \frac{n(n + 1)}{2} + ∏_{i=1}^{10^{1000000000000000000}}(n − i) .

Empirical Proof: True for 0 < n ≤ 10^{1000000000000000000} (check!), hence true for all n. Theorem 2: For all n > 0

\frac{n}{10^{1000000000000000000}} < 1 .

Empirical Proof: True for 0 < n < 10^{1000000000000000000} (check!), hence true for all n. Definition: The Pisot sequence , E(x, y) where 0 < x < y are integers, is defined by the following nonlinear recurrence:

a_{0} = x , a_{1} = y ,

and, for n > 1,

a_{n} := ⌊ \frac{a_{n−1}^{2}}{a_{n−2}} + \frac{1}{2} ⌋ ,

where, as usual, ⌊ x ⌋ denotes the largest integer that is ≤ x.