ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 17 (2022), 287 -- 303
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A NOTE ON THE PRIMALITY OF SUMS
Antonie Dinculescu
Abstract. It is shown that when adding a large number to a set of much smaller numbers, the number of primes or twin ranks (see text) in the resulted sumset can be substantially larger than the theoretical values given by the Prime Number Theorem or Hardy-Littlewood Conjecture. Specifically, the ratios of the observed numbers of primes or twin ranks to the calculated values are proportional to the logarithm and, respectively, logarithm squared of the prime omega function of the large number. A consequence of this characteristic is the existence in the set of positive integers of an infinite number of intervals where all primes or twin ranks are separated from a certain number inside by distances that are primes or twin ranks.
2020 Mathematics Subject Classification: 11A41; 11Y11
Keywords: Prime numbers, Twin primes, Primality
References
A. Dinculescu, On the numbers that determine the distribution of twin primes, Surveys in Mathematics and its Applications 13 (2018), 171--181. Zbl 1413.11012.
G. H. Hardy and J. E. Littlewood, Some problems of "Partitio numerorum"; III: On the expression of a number as a sum of primes, Acta Matematica, 44 (1923), 1--70. MR1555183. JFM 48.0143.04.
D. J. Newman, Simple analytic proof of prime number theorem, Am. Math. Monthly 87 (1980), 693--696. MR0602825. Zbl 0444.10033.
OEIS, The On-Line Encyclopedia of Integer Sequences. https://oeis.org/.
Antonie Dinculescu
Retired; formerly with "Esperion" at Western Michigan University, USA.
4148, NW 34th Drive, Gainesville, Florida, 32605.
e-mail: antoniedinculescu@gmail.com