Displaying similar documents to “The Fibonacci sequence and the golden quadratic”

An arithmetic formula of Liouville

Erin McAfee, Kenneth S. Williams (2006)

Journal de Théorie des Nombres de Bordeaux

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An elementary proof is given of an arithmetic formula, which was stated but not proved by Liouville. An application of this formula yields a formula for the number of representations of a positive integer as the sum of twelve triangular numbers.

Five regular or nearly-regular ternary quadratic forms

William C. Jagy (1996)

Acta Arithmetica

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1. Introduction. In a recent article [6], the positive definite ternary quadratic forms that can possibly represent all odd positive integers were found. There are only twenty-three such forms (up to equivalence). Of these, the first nineteen were proven to represent all odd numbers. The next four are listed as "candidates". The aim of the present paper is to show that one of the candidate forms h = x² + 3y² + 11z² + xy + 7yz does represent all odd (positive) integers, and that it is...