On the simple normality to base 2 of the square root of s, for s not a perfect square

TL;DR

This paper proves that the binary expansion of the square root of any non-perfect square integer is simply normal to base 2, using elementary finite difference calculus.

Contribution

It provides a new elementary proof that the square root of non-perfect squares are simply normal in base 2, a result previously unestablished.

Findings

Square roots of non-perfect squares are simply normal in base 2.

Elementary calculus of finite differences can be used to prove normality.

The proof advances understanding of normality for algebraic irrationals.

Abstract

We show that each number of the form, the square root of s for s not a perfect square, is simply normal to the base 2. The argument uses some elementary ideas from the calculus of finite differences.