Cognitive Visualization of Some Integer - Valued Polynomials

Emiliano Ippoliti

Visualization is a powerful device in scientific discovery and in conjecture's formulation and it has an increasing number of techniques and applications, revealing its cognitive and heuristic power. A.A. Zenkin [Zenkin 1991] has found an interesting new propriety of square numbers, 1, 4, 9, 25,..., f(x) = x2, x ÎN, using the so called phytograms, a cognitive and interactive visualization of number-theoretical proprieties. In this paper I am going to show how such new graphical representation of number-theoretical objects can lead to find some new proprieties and formulate two analytical conjectures. It will be shown, in particular, how visualization is essential to find new proprieties about some natural numbers series generated by integer-valued polynomials, i.e. f(x) = ax2 a bx a c, where x, f(x), a, b, cÎN, and f(x) = xn, i.e. f(x) = x2n+1 and f(x) = x2n where x,n,f(x), n ÎN. In §1 I will give a short introduction and will briefly describe Zenkin's result and his visualization's devices (CCG - Cognitive Computer Graphics). In §2 and §3 we will present our visual results and conjectures about the above polynomials.