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.
- Prime1 x Prime2 = SemiPrime
- arithmetically and conventionally speaking;
- literally HALF a prime, whereas the other term Biprime means more ‘dual prime’;
- Vertical Primei x Horizontal Primej = SemiPrimeij below the diagonal of cellsii
- “digitally” speaking:
- the Last Digit determines the type of number by the resulting digital value: odd or even;
- Horizontal Primei x Vertical Primej creates SemiPrimesij below the diagonal for i < j
- Horizontal Primei x Vertical Primej creates SemiPrimesij above the diagonal for i > j
- symmetrically speaking with respect to the diagonal line of cellsii
- 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 Rowi x a cell in Columnj = Cellij in ‘geometric position’ ij
- Cellji = a cell in Rowj x a cell in Columni in a 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