Proof of Luck

Abstract

A simple proof confirms Riemann, Generalized Riemann, Collatz, Swinnerton-Dyer conjectures, and Fermat's Last Theorem.

Analytic Error Function and Numeric Inverse Obtained by Geometric Means

“…As the latter was the power series expansion of (2/π)e −2t 2 , which was convergent for all values of t, the original series was then also convergent and, thus, erf p (t) 2 with the limiting value shown in Equation (7). A more compact form of the power series expansion was expressed by the following:…”

“…Based on the geometric approach described in [7], we were able to describe simple, useful formulas that, when guided by consistently higher orders of the approximation (2) for the error function, led to consistently more advanced approximations of the inverse error function. The starting point was the degree p = 0, that is, the approximation in Equation (3).…”

“…Since then, there have been a couple of derivations of this formula [4][5][6]. In Section 2, we describe an additional one that is based on the same geometric considerations as employed in [7]. In Section 3, we provide the series expansion for Craig's integral representation and show the rapid convergence of this series.…”

“…Translating the results of [7] into the error function, we obtained the approximation of order p by the following:…”

“…, N) are found in the intervals between 1/ cos(π(n − 1)/(4N)) and 1/ cos(πn/(4N)). A method for selecting those values was extensively described in [7], where the authors showed the following:…”