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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The prime number graph

Author: Carl Pomerance
Journal: Math. Comp. 33 (1979), 399-408
MSC: Primary 10A25; Secondary 52A10
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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Article copyright: © Copyright 1979 American Mathematical Society