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arXiv · 1904.03573

SanD primes and numbers

Freeman J. Dyson·Norman E. Frankel·Anthony J. Guttmann
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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.

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Freeman J. Dyson, Norman E. Frankel, Anthony J. Guttmann. 2019-04-07. SanD primes and numbers. https://arxiv.org/abs/1904.03573

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