### A combinatorial proof of a result from number theory.

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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.

My recent book Antieigenvalue Analysis, World-Scientific, 2012, presented the theory of antieigenvalues from its inception in 1966 up to 2010, and its applications within those forty-five years to Numerical Analysis, Wavelets, Statistics, Quantum Mechanics, Finance, and Optimization. Here I am able to offer three further areas of application: Continuum Mechanics, Economics, and Number Theory. In particular, the critical angle of repose in a continuum model of granular materials is shown to be exactly...

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine ${q}^{[p/8]}\left(modp\right)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.