**To explore **the intimate ‘pair relationship’ between primes and SemiPrimes, we need to differentiate not just between *sums* and *products, *but also between writing* *algebraic equations and coding them as well as using *software *such as spreadsheets for visualising* *

- the
**Triangle**of SemiPrimes - the
**Matrix**of SemiPrimes.

**Definitions**

- Prime
**1**x Prime**2**= SemiPrime*arithmetically*and*conventionally*speaking;*literally*HALF a prime, whereas the other term Biprime means more ‘dual prime’;

*Vertical*Primex_{i}*Horizontal*Prime= SemiPrime_{j}below the_{ij}**diagonal**of**cells**_{ii}*“digitally”*speaking:- the
*Last Digit*determines the*type*of number by the resulting*digital*value: odd or even;

*Horizontal*Primex_{i}*Vertical*Primecreates SemiPrimes_{j}below the diagonal for i < j_{ij}*Horizontal*Primex_{i}*Vertical*Primecreates SemiPrimes_{j}above the diagonal for i > j_{ij}*symmetrically*speaking with respect to the diagonal line of cells_{ii}*geometrically*speaking with respect to the*horizontal*axis as reference.

**Methods of Generalisation**

It is fair to say that *spreadsheet indexing* provides more ‘operational scope’ than the algebraic definition. Spreadsheet indexing can, of course, be *generalised *by *coding *up to values of i and j that go far beyond spreadsheet sizes.

This is the critical difference between *algebra* and *coding* as method of generalisation.

**In terms of Functional Matrices:**

- a cell in Row
x a cell in Column_{i}= Cell_{j}_{ij}in*‘geometric position’***ij** - Cell
_{ji}= a cell in Rowx a cell in Column_{j}in a_{i}*Diagonally Symmetrical*Matrix of**i x j**Products, i.e. a*2D area*of cells - and finally we consider Last Digit combinations as Factors for Products as
*numerical*results:

Since all Primes are ODD, only odd SemiPrimes can be the result.

Collection 9: The Factorisation of SemiPrimes shows this gallery of screenshots:

NEXT: 4. Functional Matrices

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