The prime number graph
Author:
Carl Pomerance
Journal:
Math. Comp. 33 (1979), 399408
MSC:
Primary 10A25; Secondary 52A10
DOI:
https://doi.org/10.1090/S00255718197905148367
MathSciNet review:
514836
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Abstract: Let ${p_n}$ denote the nth prime. The prime number graph is the set of lattice points $(n,{p_n})$, $n = 1,2, \ldots$. We show that for every k there are k such points that are collinear. By considering the convex hull of the prime number graph, we show that there are infinitely many n such that $2{p_n} < {p_{n  i}} + {p_{n + i}}$ for all positive $i < n$. By a similar argument, we show that there are infinitely many n for which $p_n^2 > {p_{n  i}}{p_{n + i}}$ for all positive $i < n$, thus verifying a conjecture of Selfridge. We make some new conjectures.

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© Copyright 1979
American Mathematical Society