Wagstaff, Samuel S. jun. Prime numbers with a fixed number of one bits or zero bits in their binary representation. (English) Zbl 1046.11089 Exp. Math. 10, No. 2, 267-273 (2001). The paper starts with several historical examples which suggest that proving any general result on the distribution of zero and one bits in the binary representation of prime numbers is a hard task. Thus for instance, it is not proved (yet believed to be true) that the number of primes \(p < x\) with an odd (or even) number of one bits is asymptotically \(\pi(x)/2\).The author then concentrates on the question of determining asymptotics for the number of primes with a fixed (small) number of one or zero bits in their binary expansion. Let \(P_{k}, Q_{k}\) denote the sets of all the primes with exactly \(k\) one bits, resp. \(k\) zero bits in the binary expansion; the author considers the functions \[ A_{k}(n) = \left| P_{k} \cap \left[ 2^{n}, 2^{n+1} \right]\right| \quad \hbox{ and } \quad B_{k}(n) = \left| Q_{k} \cap \left[ 2^{n}, 2^{n+1} \right]\right|. \] First he derives some raw heuristics for these functions and then the heuristics are compared to experimental values for \(n \leq 200\) and \(k=3, 4\), resp. \(k = 1, 2\). Heuristics and experiment do not match the expectation, and the author investigates several phenomena which may explain some of the divergences. The paper concludes with some obvious open questions. Reviewer: Preda Mihailescu (Paderborn) Cited in 2 Documents MSC: 11Y11 Primality 11A41 Primes 11A67 Other number representations × Cite Format Result Cite Review PDF Full Text: DOI Euclid EuDML Online Encyclopedia of Integer Sequences: Smallest prime with Hamming weight n (i.e., with exactly n 1’s when written in binary). Number of primes with three 1-bits (A081091) in range [2^n,2^(n+1)]. Number of primes with four 1-bits (A095077) in range ]2^n,2^(n+1)]. Number of primes with two 0-bits (A095079) in range ]2^n,2^(n+1)]. References: [1] Baillie R., Math. Comp. 33 pp 1333– (1979) [2] B’esineau J., C. R. Acad. Sci. Paris Sér. A-B 272 pp A453– (1971) [3] Brillhart J., Factorizations of bn {\(\pm\)}1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers, (1988) [4] Dartyge C., J. Number Theory 81 pp 270– (2000) · Zbl 0957.11039 · doi:10.1006/jnth.1999.2458 [5] Fouvry E., Math. Annalen 305 pp 571– (1996) · Zbl 0858.11050 · doi:10.1007/BF01444238 [6] Gel’fond A. O., Acta Arith. 13 pp 259– (1967) [7] Hardy G. H., An Introduction to the Theory of Numbers, (1960) · Zbl 0086.25803 [8] Hooley C., J. Reine Angew. Math. 225 pp 209– (1967) [9] Lehmer D. H., Studies in Mathematical Analysis and Related Topics, IV, Essays in Honor of George Pólya pp 202– (1962) [10] Montgomery H. L., Ten lectures on the interface between analytic number theory and harmonic analysis (1994) · Zbl 0814.11001 [11] Olivier M., C. R. Acad. Sci. Paris S’er. A-B 272 pp A937– (1971) [12] Pohlig S., IEEE Trans, on Info. Theory 24 (1) pp 106– (1978) · Zbl 0375.68023 · doi:10.1109/TIT.1978.1055817 [13] Stolarsky K. B., Acta Arith. 38 pp 117– (1980) [14] Wagstaff S. S., Math. Comp. 40 pp 385– (1983) · doi:10.1090/S0025-5718-1983-0679454-X This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.