q-series, elliptic curves, and odd values of the partition function
Let p(n) be the number of partitions of an integer n. Euler proved the following recurrence for p(n): p(n)=∑k=1∞(−1)k+1(p(n−ω(k))+p(n−ω(−k))), (*) where ω(k)=(3k 2+k)/2. In view of Euler's result, one sees that it is fairly easy to compute p(n) very quickl...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
1999-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171299220558 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849406056964292608 |
|---|---|
| author | Nicholas Eriksson |
| author_facet | Nicholas Eriksson |
| author_sort | Nicholas Eriksson |
| collection | DOAJ |
| description | Let p(n) be the number of partitions of an integer n. Euler proved the following recurrence for p(n): p(n)=∑k=1∞(−1)k+1(p(n−ω(k))+p(n−ω(−k))), (*) where ω(k)=(3k 2+k)/2. In view of Euler's result, one sees that it is fairly easy to compute p(n) very quickly. However, many questions remain open even regarding the parity of p(n). In this paper, we use various facts about elliptic curves and q-series to construct, for every i≥1, finite sets Mi for which p(n) is odd for an odd number of n∈Mi. |
| format | Article |
| id | doaj-art-5264d5bb4490487c885940f805e552d3 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1999-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-5264d5bb4490487c885940f805e552d32025-08-20T03:36:31ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251999-01-01221556510.1155/S0161171299220558q-series, elliptic curves, and odd values of the partition functionNicholas Eriksson02401 S. Hills Dr., Missoula, MT 59803, USALet p(n) be the number of partitions of an integer n. Euler proved the following recurrence for p(n): p(n)=∑k=1∞(−1)k+1(p(n−ω(k))+p(n−ω(−k))), (*) where ω(k)=(3k 2+k)/2. In view of Euler's result, one sees that it is fairly easy to compute p(n) very quickly. However, many questions remain open even regarding the parity of p(n). In this paper, we use various facts about elliptic curves and q-series to construct, for every i≥1, finite sets Mi for which p(n) is odd for an odd number of n∈Mi.http://dx.doi.org/10.1155/S0161171299220558 |
| spellingShingle | Nicholas Eriksson q-series, elliptic curves, and odd values of the partition function International Journal of Mathematics and Mathematical Sciences |
| title | q-series, elliptic curves, and odd values of the partition function |
| title_full | q-series, elliptic curves, and odd values of the partition function |
| title_fullStr | q-series, elliptic curves, and odd values of the partition function |
| title_full_unstemmed | q-series, elliptic curves, and odd values of the partition function |
| title_short | q-series, elliptic curves, and odd values of the partition function |
| title_sort | q series elliptic curves and odd values of the partition function |
| url | http://dx.doi.org/10.1155/S0161171299220558 |
| work_keys_str_mv | AT nicholaseriksson qseriesellipticcurvesandoddvaluesofthepartitionfunction |