# zbMATH — the first resource for mathematics

The Gaussian zoo. (English) Zbl 1045.11071
Let $$A= \{a_{1}, \dots, a_{k}\}$$ be a finite set of integers such that for every prime $$p$$ the number of distinct entries in the mod $$p$$ reduction of $$A$$ is less than $$p.$$ Then it is conjectured that there exist infinitely many integers $$n$$ such that $$n + a_{1}, \dots, n + a_{k}$$ are primes (“prime $$k$$-tuples conjecture”). Such sets $$A$$ are called admissible. The authors determine all the maximal admissible connected sets of primes in Gaussian integers. The paper also contains an analogue of Mertens formula for Gaussian integers.

##### MSC:
 11P32 Goldbach-type theorems; other additive questions involving primes 11N80 Generalized primes and integers
Full Text:
##### References:
 [1] Edwards H. M., Riemann’s zeta function (1974) · Zbl 0315.10035 [2] Friedlander J., J. Amer. Math. Soc. 4 (1) pp 25– (1991) · doi:10.1090/S0894-0347-1991-1080647-5 [3] Furry W. H., Nature 150 pp 120– (1942) · Zbl 0063.01486 · doi:10.1038/150120a0 [4] Gethner E., Experiment. Math. 6 (4) pp 289– (1997) · Zbl 1115.11318 · doi:10.1080/10586458.1997.10504616 [5] Hardman N. R., Amer. Math. Monthly 74 pp 559– (1967) · Zbl 0166.05802 · doi:10.2307/2314890 [6] Hardy G. H., Acta Math. 44 pp 1– (1922) · JFM 48.0143.04 · doi:10.1007/BF02403921 [7] Hardy G. H., An introduction to the theory of numbers, (1960) · Zbl 0086.25803 [8] Ireland K. F., A classical introduction to modern number theory, (1982) · Zbl 0482.10001 [9] Jordan J. R., Math. Mag. 38 pp 1– (1965) · Zbl 0124.02301 · doi:10.2307/2688007 [10] Jordan J. H., J. Number Theory 8 (1) pp 43– (1976) · Zbl 0333.12001 · doi:10.1016/0022-314X(76)90020-2 [11] Riesel H., Prime numbers and computer methods for factorization, (1985) · Zbl 0582.10001 · doi:10.1007/978-1-4757-1089-2 [12] Tenenbaum G., The prime numbers and their distribution (2000) · Zbl 0942.11001 · doi:10.1090/stml/006 [13] Uchiyama S., Proc. Amer. Math. Soc. 28 pp 629– (1971) [14] Vardi I., Computational recreations in Mathematica (1991) · Zbl 0786.11002 [15] Vardi I., Experiment. Math. 7 (3) pp 275– (1998) · Zbl 0926.11065 · doi:10.1080/10586458.1998.10504373 [16] Vardi I., Comm. Math. Phys. 207 (1) pp 43– (1999) · Zbl 0953.60096 · doi:10.1007/s002200050717
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.