Note that if X is allowed a value of zero, then all P are prime, and if allowed a value of one, then all P are not prime. As written, no Y can take the values of zero or one, as no combination of P and X allows for this possibility.

These statements seem to define 0 and 1 as being Prime Numbers. I think that this more formal definition makes the Primes more easily understood, but my investigations, using a spreadsheet program, leave the question of discovering more Primes unclear to me. Does this approach make the Riemann hypothesis any easier to prove? Does the resemblance to fractal systems or cellular automata say anything particularly important?

Below is a table of the differences of the absolute values. There definitely seems to be some sort of hidden pattern. For instance, the bottom two diagonals only really change if the larger primes are altered to even rather than odd values, and they don't seem to have any real value in predicting the next prime, while the row directly beneath the primes is far more interesting, and yet far less regular.

0 | 1 | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | 97 |

1 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 6 | 6 | 2 | 6 | 4 | 2 | 4 | 6 | 6 | 2 | 6 | 4 | 2 | 6 | 4 | 6 | 8 | |

0 | 0 | 1 | 0 | 2 | -2 | 2 | 2 | 0 | -4 | 4 | -2 | -2 | 2 | 2 | 0 | -4 | 4 | -2 | -2 | 4 | -2 | 2 | 2 | ||

0 | 1 | -1 | 2 | 0 | 0 | 0 | -2 | 4 | 0 | -2 | 0 | 0 | 0 | -2 | 4 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | |||

1 | 0 | 1 | -2 | 0 | 0 | 2 | 2 | -4 | 2 | -2 | 0 | 0 | 2 | 2 | -4 | 2 | -2 | 2 | 0 | -2 | 0 | ||||

-1 | 1 | 1 | -2 | 0 | 2 | 0 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | |||||

0 | 0 | 1 | -2 | 2 | -2 | 2 | 0 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | ||||||

0 | 1 | 1 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 0 | |||||||

1 | 0 | -1 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | ||||||||

-1 | 1 | -1 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | |||||||||

0 | 0 | -1 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | ||||||||||

0 | 1 | 1 | -2 | 0 | 0 | 2 | -2 | 0 | 2 | -2 | 2 | -2 | 0 | 2 | |||||||||||

1 | 0 | 1 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | ||||||||||||

-1 | 1 | 1 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 2 | 0 | |||||||||||||

0 | 0 | 1 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 2 | -2 | ||||||||||||||

0 | 1 | -1 | 0 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | |||||||||||||||

1 | 0 | -1 | 0 | 2 | -2 | 0 | 2 | 0 | -2 | ||||||||||||||||

-1 | 1 | -1 | 2 | 0 | -2 | 2 | -2 | 2 | |||||||||||||||||

0 | 0 | 1 | -2 | 2 | 0 | 0 | 0 | ||||||||||||||||||

0 | 1 | 1 | 0 | -2 | 0 | 0 | |||||||||||||||||||

1 | 0 | -1 | 2 | -2 | 0 | ||||||||||||||||||||

-1 | 1 | 1 | 0 | -2 | |||||||||||||||||||||

0 | 0 | -1 | 2 | ||||||||||||||||||||||

0 | 1 | 1 | |||||||||||||||||||||||

1 | 0 | ||||||||||||||||||||||||

-1 |

While reading "The Art of Computer Programming" by Donald Knuth, I have come across "(Fermat's Theorem, 1640)", which seems to be in error concerning the primality of zero, or at least to be problematic in handling division by zero.

UPDATE: As I currently foresee no reasonable way to have "x modulo 0" equal 1 even once, much less consistently, I believe that the only rational conclusion is that zero cannot be considered prime, except in the most technical sense: it retains the QUALITY of being prime, but it's QUANTITY of "Primality" is... ZERO!

If we compare the concept of "Primality" with "Complexity", then the Primality of an integer is roughly equivalent to its magnitude, and it's computational complexity in proving its being a Prime Number! This in turn could be useful as a benchmark of the power and complexity of various algorithms, computational methods, reactive processes, etc.

Professor Knuth, I believe you have previously sent me one of your famous checks in answer to this conjecture, but it has been lost to the slings and arrows of outrageous fortune. If you could send me another, I would be proud to display it here. Thank you!

If Fermat is correct, (HE IS) AND my conjecture is ALSO correct (MOSTLY), then there is a problem with Modulo 0, in that it must consistently equal 1.(Nope, I was wrong here.) Is there a solution? I have a study of Modulo 0 below if it is helpful to anyone. The result I show is not a proof by any means, and even if it's rough assertion is true, it still needs a great deal of work to even be worth working toward such a proof, but it offers some tantalizing possibilities to me as an informal and growing Mathematician. Any effort I put to this end will still be a tremendous learning experience, "pass" or "fail"! (SEEMS LIKE "FAIL", OH WELL.)