2. “Digital TABLES” for Organising PRIMES

Prime Numbers fit a pattern of graphically predictable “Vertical Prime Pairs” of Primes and SemiPrimes [Prime1 x Prime2] in these Digital Tables:

The numerical values are arranged by

• vertical Leading Digit x 10

• + horizontal Last Digit.

The pattern repeats every 30 numbers and even though the positions of the pairs are clearly NOT random, the values and distribution of primes and semiprimes appear not to be predictable.

Thus we see the fundamental digital differences between:

• multiplication which involves two factors
• addition which uses two summands
• and counting which adds 1 and decimal Orders of Magnitude [10⁰, 10¹, 10²…]

Furthermore, these screenshots display the sorting of Prime Numbers by their Last Digits. And they show the difference between PRIME Digits 2 3 5 7 and LAST Digits 1 3 7 9. They have 3 and 7 in common, whereas the differences are 2 and 5 vs 1 and 9:

• LAST Digits act as potential Prime “indicators“;

• PRIME Digits are of arithmetic and dimensional significance:-
  • 1 and 9:
    • as Last Digits they are 10 ± 1 as a basis for additions and subtractions;
    • additions and subtractions are linear and bi-directional or 1D operations;
  • 2 and 5:
    • as Factors 2 x 5 and 5 x 2 they form vertical and horizontal rectangles as ‘decimal units’, geometrically speaking;
    • multiplication is square or rectangular as 2D operation.

Below 10:

• Prime Digits 2 and 5 form 2 x 5 = 10, i.e. they lay the foundation for 2D “decimal rectangles“, geometrically speaking.

Above 10:

• Last Digits 1 and 9 = 10 ± 1, i.e. they surround multiples of 10 along the 1D Number Line.

When Prime Numbers become ‘Prime Factors’ to produce SemiPrimes, we see these tables:

Why on earth could these series be important in this pattern? Let me count the reasons:

1. in terms of education and the difference between belief vs knowledge, Plato’s analogy of the Cave tells us that we need to distinguish between:
   • what is in our mind from the use of words as concepts
   • and what is before our eyes from the use of images as perceptions;
2. the general assumption is that there is no pattern, and thus no predictability or formula for Prime Numbers;
   • these “Digital Table” patterns predict the presence of “Vertical Pairs” of Primes (sums) and SemiPrimes (products) in specific cell positions;
   • this means a digital or ‘holy mix’ between sums and products;
3. equally ‘difficult’ is the Factorisation of SemiPrimes described as the ‘hardest’ on this Wikipedia page on Integer factorisation;
   • yet ‘deep’ and ‘spreadsheet-enabled’ thinking made it possible for me;
4. in my manual work the pattern was always right!
   • this meant I could predict the next product of primes for the empty fields in the pattern of “Diagonally Staggered Prime Pairs”:
     • the pattern tells me which number it will be;
     • I just had to find its factors;
5. I wrote a Factorisation Table built on the sequence 6n ± 1 which I picked up from Dr Peter Plichta in an article he published in the German magazine Spektrum in the 90s;
   • more recently, I created six screenshots as Collection 9 on The Factorisation of SemiPrimes;
6. products advance faster than sums;
   • the ‘primality’ test of whether a number is prime or not is thus always lagging behind the SemiPrimes whose prime factors are known;
   • if we turn them into a LookUp Table, Bob’s our uncle!
7. spreadsheets allow me to cut and paste much larger matrices and tables – often one cell at a time – and, again, the pattern is always right!
   • the primes are unpredictable ‘givens’;
   • SemiPrimes are ‘predictable fillers’ – most intriguingly made up from these unpredictable prime numbers;
   • the predictability of SemiPrimes is shown in a Functional Matrix:
     • i.e. the multiplication of vertical with horizontal Primes below the diagonal
     • or horizontal Primes x vertical Primes for SemiPrimes symmetrically above the diagonal.

It depends entirely on ‘prior knowledge’ or bias and preconceived ideas whether and how this website will make a difference. I’ve just tried my best to be as clear as possible.

NEXT: 3. Primes as “Diagonal Sums”