Karaji's L-Summing Method

Hossein Teimoori Faal

Abstract

Karaji was a 10th century Persian Mathematician and Engineer. His enduring contributions to the field of mathematics and engineering was still recognized today in the form of the table of Binomial coefficients. He was also the first to use the geometric idea to prove the sums of cubes identity, namely

1^3+2^3+\cdots+n^3 = (1+2+\cdots+n)^2.

Motivated by his original work, we will use it in a more algebraic setting to give a simple method for proving several classes of Algebraic Identities. We will call it Karaji's L-Summing Method. Indeed, by combining the idea of Multiplication table with Karaji's L-Summing Method, we derive several classes of identities, including:

1) Analytic Number Theory identities (Zeta function and Harmonic numbers) 2) combinatorial identities (Derangments and