Apparition and Periodicity Properties of Equianharmonic Divisibility Sequences

Citation

Durst, Lincoln Kearney (1952) Apparition and Periodicity Properties of Equianharmonic Divisibility Sequences. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4093-GY48. https://resolver.caltech.edu/CaltechTHESIS:10122017-094936829

Abstract

Elliptic divisibility sequences were first studied by Morgan Ward, who proved that they admit every prime p as a divisor and gave the upper bound 2p + 1 for the smallest place of apparition of p. He also proved that, except for a few special primes, the sequences are numerically periodic modulo p.

This thesis contains a discussion of equanharmonic divisibility sequences and mappings. These sequences are the special elliptic sequences which occur when the elliptic functions involved degenerate into equianharmonic functions, and the divisibility mappings are an extensioin of the notion of a sequence to a function over a certain ring of quadratic integers

For equianharmonic divisibility sequences and mappings an arithmetical relation between any rational prime of the form 3k + 2 and its rank of apparition is found.

It is also shown that, except for a few special prime ideals, equianharmonic divisibility mappings are numerically doubly periodic to prime ideal moduli.

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