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Greatest of the least primes in arithmetic progressions having a given modulus


Author: Samuel S. Wagstaff
Journal: Math. Comp. 33 (1979), 1073-1080
MSC: Primary 10H20; Secondary 10-04
MathSciNet review: 528061
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Abstract: We give a heuristic argument, supported by numerical evidence, which suggests that the maximum, taken over the reduced residue classes modulo k, of the least prime in the class, is usually about $ \phi (k)\log k\log \phi (k)$, where $ \phi $ is Euler's phi-function.


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  • [1] P. Erdös, On some applications of Brun’s method, Acta Univ. Szeged. Sect. Sci. Math. 13 (1949), 57–63. MR 0029941
  • [2] D. R. Heath-Brown, Almost-primes in arithmetic progressions and short intervals, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 3, 357–375. MR 0491558
  • [3] Hans-Joachim Kanold, Über Primzahlen in arithmetischen Folgen, Math. Ann. 156 (1964), 393–395 (German). MR 0169827
  • [4] E. LANDAU, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, Teubner, Leipzig-Berlin, 1909. Reprinted by Chelsea, New York, 1953.
  • [5] U. V. Linnik, On the least prime in an arithmetic progression. I. The basic theorem, Rec. Math. [Mat. Sbornik] N.S. 15(57) (1944), 139–178 (English, with Russian summary). MR 0012111
  • [6] M. E. Low, Real zeros of the Dedekind zeta function of an imaginary quadratic field, Acta Arith 14 (1967/1968), 117–140. MR 0236127
  • [7] Carl Pomerance, A note on the least prime in an arithmetic progression, J. Number Theory 12 (1980), no. 2, 218–223. MR 578815, 10.1016/0022-314X(80)90056-6
  • [8] K. Prachar, Über die kleinste Primzahl einer arithmetischen Reihe, J. Reine Angew. Math. 206 (1961), 3–4 (German). MR 0125092
  • [9] G. B. PURDY, Some Extremal Problems in Geometry and the Theory of Numbers, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1972.
  • [10] Andrzej Schinzel, Remark on the paper of K. Prachar “Über die kleinste Primzahl einer arithmetischen Reihe”, J. Reine Angew. Math. 210 (1962), 121–122 (German). MR 0150115
  • [11] E. C. TITCHMARSH, "A divisor problem," Rend. Circ. Mat. Palermo, v. 54, 1930, pp. 414-429.
  • [12] P. TURÁN, "Über die Primzahlen der arithmetischen Progression," Acta Sci. Math. (Szeged), v. 8, 1936/37, pp. 226-235.
  • [13] Samuel S. Wagstaff Jr., The irregular primes to 125000, Math. Comp. 32 (1978), no. 142, 583–591. MR 0491465, 10.1090/S0025-5718-1978-0491465-4

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1979-0528061-7
Keywords: Least prime in an arithmetic progression
Article copyright: © Copyright 1979 American Mathematical Society