Free Online Productivity Tools
i2Speak
i2Symbol
i2OCR
iTex2Img
iWeb2Print
iWeb2Shot
i2Type
iPdf2Split
iPdf2Merge
i2Bopomofo
i2Arabic
i2Style
i2Image
i2PDF
iLatex2Rtf
Sci2ools

JCT

1998

1998

Let Q(N) denote the number of partitions of N into distinct parts. If ω(k) := 3k2 +k 2 , then it is well known that Q(N) + ∞X k=1 (−1)k “ Q(N − 2ω(k)) + Q(N − 2ω(−k)) ” = ( 1 if N = m(m+1) 2 0 otherwise. In this short note we start with Tunnell’s work on the ‘congruent number problem’ and show that Q(N) often satisﬁes ‘weighted’ recurrence type relations. For every N there is a relation for Q(N) which may involve a special value of an elliptic curve L-function. A positive integer D is called a ‘congruent number’ if there exists a right triangle with rational sidelengths with area D. Over the centuries there have been many investigations attempting to classify the congruent numbers, but little was known until Tunnell [T] brilliantly applied a tour de force of methods and provided a conditional solution to this problem. It turns out that a square-free integer D is not congruent if the coeﬃcient of qD in a certain power series is non-zero, and assuming t...

Added |
22 Dec 2010 |

Updated |
22 Dec 2010 |

Type |
Journal |

Year |
1998 |

Where |
JCT |

Authors |
Ken Ono |

Comments (0)