Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur befor 1012
Let πb,c(x) denote the number of primes ≤x and ≡c(modb), and for positive integers x let Δb(x,c,l)=πb,c(x)−πb,l(x). Negative values of Δ4(x,3,1) less than 1012 occur in six widely spaced regions. The first three regions, investigated by Leech [6], Shanks [9] and Lehmer [6 ], contain only a few thous...
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Wiley
1979-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171279000119 |
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| author | Carter Bays Richard H. Hudson |
| author_facet | Carter Bays Richard H. Hudson |
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| description | Let πb,c(x) denote the number of primes ≤x and ≡c(modb), and for positive integers x let Δb(x,c,l)=πb,c(x)−πb,l(x). Negative values of Δ4(x,3,1) less than 1012 occur in six widely spaced regions. The first three regions, investigated by Leech [6], Shanks [9] and Lehmer [6 ], contain only a few thousand negative values of Δ4(x,3,1). However, the authors [1] have recently discovered 3 new regions, the sixth occurring before 20 billion and containing more than half a billion negative values of Δ4(x,3,1). In this paper numerical and graphical details of all six regions are given. Moreover, new results for the modulus 8 are presented. Previously, no negative values have been found for Δ8(x,c,1), c=3,5, or 7 and our search to 1012 reveals no such values for Δ8(x,3,1) or Δ8(x,7,1). For Δ8(x,5,1) we have discovered the first two regions of negative values. The first of these regions, beginning at x=588067889, contains 422,500 negative values of Δ8(x,5,1); the second occurs in the vicinity of 35 billion and contains more than a billion negative values of Δ8(x,5,1). |
| format | Article |
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| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1979-01-01 |
| publisher | Wiley |
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| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-ad7c6b65f53e4ab7a111ac4068e5eec02025-02-03T01:26:46ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251979-01-012111111910.1155/S0161171279000119Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur befor 1012Carter Bays0Richard H. Hudson1Department of Mathematics and Computer Science, University of South Carolina, Columbia 29208, S.C., USADepartment of Mathematics and Computer Science, University of South Carolina, Columbia 29208, S.C., USALet πb,c(x) denote the number of primes ≤x and ≡c(modb), and for positive integers x let Δb(x,c,l)=πb,c(x)−πb,l(x). Negative values of Δ4(x,3,1) less than 1012 occur in six widely spaced regions. The first three regions, investigated by Leech [6], Shanks [9] and Lehmer [6 ], contain only a few thousand negative values of Δ4(x,3,1). However, the authors [1] have recently discovered 3 new regions, the sixth occurring before 20 billion and containing more than half a billion negative values of Δ4(x,3,1). In this paper numerical and graphical details of all six regions are given. Moreover, new results for the modulus 8 are presented. Previously, no negative values have been found for Δ8(x,c,1), c=3,5, or 7 and our search to 1012 reveals no such values for Δ8(x,3,1) or Δ8(x,7,1). For Δ8(x,5,1) we have discovered the first two regions of negative values. The first of these regions, beginning at x=588067889, contains 422,500 negative values of Δ8(x,5,1); the second occurs in the vicinity of 35 billion and contains more than a billion negative values of Δ8(x,5,1).http://dx.doi.org/10.1155/S0161171279000119prime numbersquadratic non-residueand axis crossing regions. |
| spellingShingle | Carter Bays Richard H. Hudson Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur befor 1012 International Journal of Mathematics and Mathematical Sciences prime numbers quadratic non-residue and axis crossing regions. |
| title | Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur befor 1012 |
| title_full | Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur befor 1012 |
| title_fullStr | Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur befor 1012 |
| title_full_unstemmed | Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur befor 1012 |
| title_short | Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur befor 1012 |
| title_sort | numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur befor 1012 |
| topic | prime numbers quadratic non-residue and axis crossing regions. |
| url | http://dx.doi.org/10.1155/S0161171279000119 |
| work_keys_str_mv | AT carterbays numericalandgraphicaldescriptionofallaxiscrossingregionsformoduli4and8whichoccurbefor1012 AT richardhhudson numericalandgraphicaldescriptionofallaxiscrossingregionsformoduli4and8whichoccurbefor1012 |