A new formulation of the law of octic reciprocity for primes ≡±3(mod8) and its consequences
Let p and q be odd primes with q≡±3(mod8), p≡1(mod8)=a2+b2=c2+d2 and with the signs of a and c chosen so that a≡c≡1(mod4). In this paper we show step-by-step how to easily obtain for large q necessary and sufficient criteria to have (−1(q−1)/2q(p−1)/8≡(a−b)d/ac)j(modp) for j=1,…,8 (the cases with j...
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Wiley
1982-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171282000532 |
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| author | Richard H. Hudson Kenneth S. Williams |
| author_facet | Richard H. Hudson Kenneth S. Williams |
| author_sort | Richard H. Hudson |
| collection | DOAJ |
| description | Let p and q be odd primes with q≡±3(mod8), p≡1(mod8)=a2+b2=c2+d2 and with the signs of a and c chosen so that a≡c≡1(mod4). In this paper we show step-by-step how to easily obtain for large q necessary and sufficient criteria to have (−1(q−1)/2q(p−1)/8≡(a−b)d/ac)j(modp) for j=1,…,8 (the cases with j odd have been treated only recently [3] in connection with the sign ambiguity in Jacobsthal sums of order 4. This is accomplished by breaking the formula of A.E. Western into three distinct parts involving two polynomials and a Legendre symbol; the latter condition restricts the validity of the method presented in section 2 to primes q≡3(mod8) and significant modification is needed to obtain similar results for q≡±1(mod8). Only recently the author has completely resolved the case q≡5(mod8), j=1,…,8 and a sketch of the method appears in the closing section of this paper. |
| format | Article |
| id | doaj-art-03e2c166c01e42f2ae3d5ff2a286e7d4 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1982-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-03e2c166c01e42f2ae3d5ff2a286e7d42025-02-03T07:25:55ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251982-01-015356558410.1155/S0161171282000532A new formulation of the law of octic reciprocity for primes ≡±3(mod8) and its consequencesRichard H. Hudson0Kenneth S. Williams1Department of Mathematics, Carleton University, Ottawa KIS 5B6, CanadaDepartment of Mathematics, Carleton University, Ottawa KIS 5B6, CanadaLet p and q be odd primes with q≡±3(mod8), p≡1(mod8)=a2+b2=c2+d2 and with the signs of a and c chosen so that a≡c≡1(mod4). In this paper we show step-by-step how to easily obtain for large q necessary and sufficient criteria to have (−1(q−1)/2q(p−1)/8≡(a−b)d/ac)j(modp) for j=1,…,8 (the cases with j odd have been treated only recently [3] in connection with the sign ambiguity in Jacobsthal sums of order 4. This is accomplished by breaking the formula of A.E. Western into three distinct parts involving two polynomials and a Legendre symbol; the latter condition restricts the validity of the method presented in section 2 to primes q≡3(mod8) and significant modification is needed to obtain similar results for q≡±1(mod8). Only recently the author has completely resolved the case q≡5(mod8), j=1,…,8 and a sketch of the method appears in the closing section of this paper.http://dx.doi.org/10.1155/S0161171282000532quartic and octic residuacity criteriaA.E. Western's formulabinary quadratic forms. |
| spellingShingle | Richard H. Hudson Kenneth S. Williams A new formulation of the law of octic reciprocity for primes ≡±3(mod8) and its consequences International Journal of Mathematics and Mathematical Sciences quartic and octic residuacity criteria A.E. Western's formula binary quadratic forms. |
| title | A new formulation of the law of octic reciprocity for primes ≡±3(mod8) and its consequences |
| title_full | A new formulation of the law of octic reciprocity for primes ≡±3(mod8) and its consequences |
| title_fullStr | A new formulation of the law of octic reciprocity for primes ≡±3(mod8) and its consequences |
| title_full_unstemmed | A new formulation of the law of octic reciprocity for primes ≡±3(mod8) and its consequences |
| title_short | A new formulation of the law of octic reciprocity for primes ≡±3(mod8) and its consequences |
| title_sort | new formulation of the law of octic reciprocity for primes ≡ 3 mod8 and its consequences |
| topic | quartic and octic residuacity criteria A.E. Western's formula binary quadratic forms. |
| url | http://dx.doi.org/10.1155/S0161171282000532 |
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