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Mathematics of Computation

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Iterated absolute values of differences of consecutive primes

Author: Andrew M. Odlyzko
Journal: Math. Comp. 61 (1993), 373-380
MSC: Primary 11Y35; Secondary 11N05
MathSciNet review: 1182247
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Abstract: Let $ {d_0}(n) = {p_n}$, the nth prime, for $ n \geq 1$, and let $ {d_{k + 1}}(n) = \vert{d_k}(n) - {d_k}(n + 1)\vert$ for $ k \geq 0,n \geq 1$. A well-known conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that $ {d_k}(1) = 1$ for all $ k \geq 1$. This paper reports on a computation that verified this conjecture for $ k \leq \pi ({10^{13}}) \approx 3 \times {10^{11}}$. It also discusses the evidence and the heuristics about this conjecture. It is very likely that similar conjectures are also valid for many other integer sequences.

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Article copyright: © Copyright 1993 American Mathematical Society

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