SanD primes and numbers

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.

22 pages, 2 figures. Revised version corrects one proof and fixes some typos