Fast computation of Andrews’ smallest part statistic and conjectured congruences
| Autor: | Olaf Hall-Holt, Charles McEachern, Kristina C. Garrett, Todd Frederick |
|---|---|
| Rok vydání: | 2011 |
| Předmět: |
Mathematics::Number Theory
Computation Applied Mathematics Astrophysics::Cosmology and Extragalactic Astrophysics Congruence relation Ramanujan's sum Moduli Quantitative Biology::Subcellular Processes Combinatorics symbols.namesake TheoryofComputation_LOGICSANDMEANINGSOFPROGRAMS ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION symbols Partition (number theory) Discrete Mathematics and Combinatorics Time complexity Statistic Mathematics |
| Zdroj: | Discrete Applied Mathematics. 159(13):1377-1380 |
| ISSN: | 0166-218X |
| DOI: | 10.1016/j.dam.2011.04.022 |
| Popis: | Let spt ( n ) denote Andrews’ smallest part statistic. Andrews discovered congruences for spt ( n ) mod 5 , 7 and 13 which are reminiscent of Ramanujan’s classical partition congruences for moduli 5, 7, and 11. We create an algorithm exploiting a recursive pattern in Andrews’ smallest part statistic, spt ( n ) , to generate modular residues of spt values in quadratic time and linear working memory. We use this algorithm to acquire the first million values of spt ( n ) . On the basis of the data, we make conjectures about the existence of hundreds of thousands of new congruences including a simple modulus 11 congruence that was found and proved independently by Garvan. |
| Databáze: | OpenAIRE |
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