On the representation of primes by binary quadratic forms, and elliptic curves
It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of an elliptic curve over a finite field $\mathbb F_q$. F...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Published: |
Pushpa Publishing House
2018
|
| _version_ | 1826267812384997376 |
|---|---|
| author | Elia, M Pintore, F |
| author_facet | Elia, M Pintore, F |
| author_sort | Elia, M |
| collection | OXFORD |
| description | It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of an elliptic curve over a finite field $\mathbb F_q$. Further, a method is described which computes representations of primes from reduced quadratic forms by means of the integral roots of polynomials over $\mathbb Z$. Lastly, some progress is made on the still-unsettled general problem of deciding which primes are represented by which classes of quadratic forms of given discriminant. |
| first_indexed | 2024-03-06T20:59:57Z |
| format | Journal article |
| id | oxford-uuid:3a8e0cf3-0a46-4bae-8351-4f3dc33f3f99 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:59:57Z |
| publishDate | 2018 |
| publisher | Pushpa Publishing House |
| record_format | dspace |
| spelling | oxford-uuid:3a8e0cf3-0a46-4bae-8351-4f3dc33f3f992022-03-26T14:02:17ZOn the representation of primes by binary quadratic forms, and elliptic curvesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:3a8e0cf3-0a46-4bae-8351-4f3dc33f3f99Symplectic Elements at OxfordPushpa Publishing House2018Elia, MPintore, FIt is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of an elliptic curve over a finite field $\mathbb F_q$. Further, a method is described which computes representations of primes from reduced quadratic forms by means of the integral roots of polynomials over $\mathbb Z$. Lastly, some progress is made on the still-unsettled general problem of deciding which primes are represented by which classes of quadratic forms of given discriminant. |
| spellingShingle | Elia, M Pintore, F On the representation of primes by binary quadratic forms, and elliptic curves |
| title | On the representation of primes by binary quadratic forms, and elliptic curves |
| title_full | On the representation of primes by binary quadratic forms, and elliptic curves |
| title_fullStr | On the representation of primes by binary quadratic forms, and elliptic curves |
| title_full_unstemmed | On the representation of primes by binary quadratic forms, and elliptic curves |
| title_short | On the representation of primes by binary quadratic forms, and elliptic curves |
| title_sort | on the representation of primes by binary quadratic forms and elliptic curves |
| work_keys_str_mv | AT eliam ontherepresentationofprimesbybinaryquadraticformsandellipticcurves AT pintoref ontherepresentationofprimesbybinaryquadraticformsandellipticcurves |