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

HTML articles powered by AMS MathViewer

- by Carter Bays and Richard H. Hudson PDF
- Math. Comp.
**32**(1978), 571-576 Request permission

## 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

- Carter Bays and Richard H. Hudson,
*On the fluctuations of Littlewood for primes of the form $4n\not =1$*, Math. Comp.**32**(1978), no. 141, 281–286. MR**476615**, DOI 10.1090/S0025-5718-1978-0476615-8 - Carter Bays and Richard H. Hudson,
*Numerical and graphical description of all axis crossing regions for the moduli $4$ and $8$ which occur before $10^{12}$*, Internat. J. Math. Math. Sci.**2**(1979), no. 1, 111–119. MR**529694**, DOI 10.1155/S0161171279000119 - Jan Bohman,
*On the number of primes less than a given limit*, Nordisk Tidskr. Informationsbehandling (BIT)**12**(1972), 576–578. MR**321890**, DOI 10.1007/bf01932967
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 $4n \pm 3$," - A. E. Ingham,
*The distribution of prime numbers*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 30, Stechert-Hafner, Inc., New York, 1964. MR**0184920** - S. Knapowski and P. Turán,
*Further developments in the comparative prime-number theory. I*, Acta Arith.**9**(1964), 23–40. MR**162771**, DOI 10.4064/aa-9-1-23-40 - John Leech,
*Note on the distribution of prime numbers*, J. London Math. Soc.**32**(1957), 56–58. MR**83001**, DOI 10.1112/jlms/s1-32.1.56 - R. Sherman Lehman,
*On the difference $\pi (x)-\textrm {li}(x)$*, Acta Arith.**11**(1966), 397–410. MR**202686**, DOI 10.4064/aa-11-4-397-410
E. LITTLEWOOD, "Sur la distribution des nombres premiers," - Daniel Shanks,
*Quadratic residues and the distribution of primes*, Math. Tables Aids Comput.**13**(1959), 272–284. MR**108470**, DOI 10.1090/S0025-5718-1959-0108470-8

*Bull. Cl. Phys. Acad. Imp. Sci.*, v. 11, 1853, p. 208.

*Comptes Rendus*, v. 158, 1914, pp. 1869-1872.

## Additional Information

- © Copyright 1978 American Mathematical Society
- 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