Geodesic
Cognitive Visualization of Some Integer - Valued Polynomials
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.
Classification : 08A40 
@article{VM_2002_4_1_a4,
     author = {Emiliano Ippoliti},
     title = {Cognitive {Visualization} of {Some} {Integer} - {Valued} {Polynomials}},
     journal = {Visual Mathematics},
     publisher = {mathdoc},
     volume = {4},
     number = {1},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VM_2002_4_1_a4/}
}
TY  - JOUR
AU  - Emiliano Ippoliti
TI  - Cognitive Visualization of Some Integer - Valued Polynomials
JO  - Visual Mathematics
PY  - 2002
VL  - 4
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VM_2002_4_1_a4/
LA  - en
ID  - VM_2002_4_1_a4
ER  - 
%0 Journal Article
%A Emiliano Ippoliti
%T Cognitive Visualization of Some Integer - Valued Polynomials
%J Visual Mathematics
%D 2002
%V 4
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VM_2002_4_1_a4/
%G en
%F VM_2002_4_1_a4
Emiliano Ippoliti. Cognitive Visualization of Some Integer - Valued Polynomials. Visual Mathematics, Tome 4 (2002) no. 1. http://geodesic.mathdoc.fr/item/VM_2002_4_1_a4/