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{\bf Theorem 1}: For all $n>0$
$$
\sum_{i=1}^{n} i \, = \, \frac{n(n+1)}{2} + \prod_{i=1}^{10^{1000000000000000000}} (n-i) \quad .
$$

{\bf Empirical Proof}: True for $0 <n \leq 10^{1000000000000000000}$ (check!), hence true for all $n$.

{\bf Theorem 2}: For all $n>0$
$$
\frac{n}{10^{1000000000000000000}} < 1 \quad .
$$

{\bf Empirical Proof}: True for $0 <n < 10^{1000000000000000000}$ (check!), hence true for all $n$.


{\bf Definition}: The Pisot sequence , $E(x,y)$  where
$0 < x < y$ are integers, is defined by the following nonlinear recurrence:
$$
a_0=x \quad , \quad a_1=y \quad ,
$$
and, for $n>1$,
$$
a_n:= \left\lfloor \, \frac{a_{n-1}^2}{a_{n-2}}  + \frac{1}{2} \, \right\rfloor \quad ,
$$
where, as usual, $\lfloor \, x \, \rfloor$ denotes the largest integer that is $\leq x$.

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