Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Details of the first region of integers $ x$ with $ \pi \sb{3,2}(x)<\pi \sb{3,1}(x)$


Authors: Carter Bays and Richard H. Hudson
Journal: Math. Comp. 32 (1978), 571-576
MSC: Primary 10-04; Secondary 10H20
DOI: https://doi.org/10.1090/S0025-5718-1978-0476616-X
MathSciNet review: 0476616
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Since the time of Chebyshev [4] there has been interest in the magnitude of the smallest integer x with $ {\pi _{3,2}}(x) < {\pi _{3,1}}(x)$, where $ {\pi _{b,c}}(x)$ denotes the number of positive primes $ \leqslant x$ and $ \equiv c\;\pmod b$. The authors have recently reached this threshold with the discovery that $ {\pi _{3,2}}(608981813029) - {\pi _{3,1}}(608981813029) = - 1$. This paper includes a detailed numerical and graphical description of values of $ {\pi _{3,2}}(x) - {\pi _{3,1}}(x)$ in the vicinity of this long sought number.


References [Enhancements On Off] (What's this?)

  • [1] CARTER BAYS & RICHARD H. HUDSON, "On the fluctuations of Littlewood for primes of the form $ 4n \pm 1$," Math. Comp., v. 32, 1978, pp. 281-286. MR 0476615 (57:16174)
  • [2] CARTER BAYS & RICHARD H. HUDSON, "Numerical and graphical description of all axis crossing regions for the moduli 4 and 8 which occur before $ {10^{12}}$." (To appear.) MR 529694 (80h:10003)
  • [3] J. BOHMAN, "On the number of primes less than a given limit," BIT, v. 12, 1972, pp. 576-577. MR 0321890 (48:255)
  • [4] P. L. CHEBYSHEV, "Lettre de M. le Professeur Tchébychev à M. Fuss sur un noveaux théorème rélatif aux nombres premiers contenus dans les formes $ 4n \pm 1$ et $ 4n \pm 3$," Bull. Cl. Phys. Acad. Imp. Sci., v. 11, 1853, p. 208.
  • [5] A. E. INGHAM, The Distribution of Prime Numbers, Cambridge Tracts in Math. and Math. Phys., no. 30, Stechert-Hafner, New York, 1964. MR 0184920 (32:2391)
  • [6] S. KNAPOWSKI & P. TURÁN, "Further developments in the comparative primenumber theory. I, "Acta Arith., v. 9, 1964, pp. 23-40. MR 0162771 (29:75)
  • [7] JOHN LEECH, "Note on the distribution of prime numbers," J. London Math. Soc. v. 32, 1957, pp. 56-58. MR 0083001 (18:642d)
  • [8] R. SHERMAN LEHMAN, "On the difference $ \pi (x) - {\text{li}}(x)$," Acta Arith., v. 11, 1966, pp. 397-410. MR 0202686 (34:2546)
  • [9 J] E. LITTLEWOOD, "Sur la distribution des nombres premiers," Comptes Rendus, v. 158, 1914, pp. 1869-1872.
  • [10] DANIEL SHANKS, "Quadratic residues and the distribution of primes," MTAC, v. 13, 1959, pp. 272-284. MR 0108470 (21:7186)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10-04, 10H20

Retrieve articles in all journals with MSC: 10-04, 10H20


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1978-0476616-X
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society