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Prime-representing functions. (English) Zbl 1240.11101
W. H. Mills [Bull. Am. Math. Soc. 53, 604 (1947; Zbl 0033.16303)] proved: There exists a real constant \(\alpha\) such that the sequence \(\lfloor \alpha^{3^n}\rfloor\) contains only prime numbers. There have been some subsequent extensions or refinements: I. Niven [Proc. Am. Math. Soc. 2, 753–755 (1951; Zbl 0044.03702)] proved that the \(3\) can be replaced by any constant \(c>\frac{8}{3}\), G. Alkauskas and A. Dubickas [Acta Math. Hung. 105, No.  3, 249–256 (2004; Zbl 1102.11004)] improved this to \(c>\frac{40}{19}\). These constants reflect the current knowledge on primes in short intervals. Along this approach, reducing the constant to 2 appeared to be hopeless. E. M. Wright [J. Lond. Math. Soc. 29, 63–71 (1954; Zbl 0055.04101)] extended the result to hold for certain sequences.
Making use of her recent result, \[ \sum_{p_{n+1}-p_n> x^{1/2},\;x\leq p_{n\leq 2x}} (p_{n+1}-p_n) \ll x^{2/3}, \] [Q. J. Math. 58, No.  4, 489–518 (2007; Zbl 1141.11042)], the author proves: Let \(c_i\) be a sequence of real numbers with \(c_i\geq 2\). Let \(C_n = c_1 \cdots c_n\). There exists \(\alpha>2\) such that the sequence \(\lfloor \alpha^{C_n} \rfloor\) contains only prime numbers. The set of such numbers \(\alpha\) has the cardinality of the continuum, is nowhere dense and has measure zero. In particular for \(c_i=2^i\) this shows that there is some \(\alpha >2\) such that \(\lfloor \alpha^{2^n} \rfloor\) contains only prime numbers. On the Riemann Hypothesis the condition can be weakened to \(c_i \geq \frac{1+\sqrt{5}}{2}\).

MSC:
11N05 Distribution of primes
11A41 Primes
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References:
[1] G. Alkauskas and A. Dubickas, Prime and composite numbers as integer parts of powers, Acta Math. Hungar., 105 (2004), 249–256. · Zbl 1102.11004
[2] R. C. Baker, G. Harman and J. Pintz, The difference between consecutive primes, II, Proc. London Math. Soc. (3), 83 (2001), 532–562. · Zbl 1016.11037
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[5] W. H. Mills, A prime representing function, Bull. Amer. Math. Soc., 53 (1947), 604. · Zbl 0033.16303
[6] I. Niven, Functions which represent prime numbers, Proc. Amer. Math. Soc., 2 (1951), 753–755. · Zbl 0044.03702
[7] A. S. Peck, On the differences between consecutive primes, PhD thesis (University of Oxford, 1996). · Zbl 0891.11046
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[9] A. Selberg, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid, 47 (1943), 87–105. · Zbl 0063.06869
[10] E. M. Wright, A class of representing functions, J. London Math. Soc., 29 (1954), 63–71. · Zbl 0055.04101
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