# SanD primes and numbers

## TL;DR

This paper introduces SanD numbers and SanD primes, analyzing their distribution and asymptotic growth rates in decimal and binary systems, revealing their quasi-fractal digital-sum properties.

## Contribution

It defines SanD numbers and primes, and derives their asymptotic growth rates, highlighting their unique digital-sum based structure and distribution patterns.

## Key findings

- SanD numbers grow linearly with constant 2/3
- SanD primes grow approximately as x / log^2 x with constant 3/4
- Digital-sum function exhibits quasi-fractal behavior affecting convergence

## Abstract

We define S(um)anD(ifference) numbers as ordered pairs $(m,\, m+\Delta)$ such that the digital-sum $DS(m(m+\Delta))=\Delta.$ We consider both the decimal and the binary case. If both $m$ and $m+\Delta$ are prime numbers, we refer to SanD {\em primes}. We show that the number of (decimal-based) SanD numbers less than $x$ grows as $c1\cdot x,$ where $c1 = 2/3,$ while the number of SanD primes less than $x$ grows as $c2\cdot x/\log^2{x},$ where $c2 = 3/4.$ Due to the quasi-fractal nature of the digital-sum function, convergence is both slow and erratic compared to twin primes, which, apart from the constant, have the same leading asymptotics.