Generic Sixteen-Squared Numbers (Seds)




The sixteen square relationship does not exist. The closest approach is Young's sixteen to 32 result, or the non-distributive approaches of Albrecht Pfister, Conway and Warren Smith.

The approach below uses an iterative solver using GiNaC as a computer algebra system to demonstrate conflicting product definitions inherent in an attempted real value only product system. This is distinct from the digital logic approach presented on the parent page.

The computation runs as follows. First, assign free variables, using the gamma pattern. (Other patterns work, as long as you don't overspecify four related elements.)


              Gamma Pattern

        a b c d e f g h i j k l m n o p
        q . . . . . . . r . . . s . t .
        u . . . . . . . v . . . w . . .
        x . . . . . . . A . . . B . . .
        C . . . . . . . D . . . . . . .
        E . . . . . . . F . . . . . . .
        G . . . . . . . H . . . . . . .
        I . . . . . . . J . . . . . . .
        K . . . . . . . . . . . . . . .
        L . . . . . . . . . . . . . . .
        M . . . . . . . . . . . . . . .
        N . . . . . . . . . . . . . . .
        O . . . . . . . . . . . . . . .
        P . . . . . . . . . . . . . . .
        Q . . . . . . . . . . . . . . .
        R . . . . . . . . . . . . . . .


In a fashion similar to QuadsViaMult.cp and OctsViaMult.cp, we then solve for sign expressions in an iterative fashion. The dependent variables are the 'x's, which are determined using the four-term relationship.
	col = rr^row^cc;
	T[rr][cc] = -T[row][cc]*T[row][col]*T[rr][col];
The program SedsViaMult.cp is developed and run in stages. Initially, we only define our independent variables. We then evaluate all definitions for the dependent variables. All discovered definitions are simplified (squared factors reduce to one), and the definitions added to the program. After sevens rounds, all cells are consistently defined. As a consistency check, each dictionary row for each cell is divided by the first column entry, and the resulting ratios evaluated with square terms reduced to one. In the case of seds, inconsistencies (1 = -1 type of errors) are rapidly found. See bottom part (line 2520) of SedsViaMult.txt.