How to Datamine for Primes Using a Spirallic Dataset

Doing Further Investigations

1. Knowing (or further discovering) the formulas for certain primes-rich rows and columns and diagonals you want to investigate, select first the left side of the line returning 10 primes and extended the series from 8 to 32 results. You'll find 11 primes in all, including the original 4 of 17,47,149 and 268, which is a result of 50% of 8. You thus obtain a new result of 11/32, or about 34%.
2. To investigate the right side of the line, follow the same procedure of analyzing the formula underlying the series 3, 2, 11, 28, 53, etc. (which you had basically already done).

   1. Extend this series by incrementing a and b by 2 each time of ((a*b)+c)+d=Sum and decrementing c by -5 while leaving d constant at 8.
   2. Try extending the series to 32 members from the 9 original ones, counting 3 as well, though it won't fit the formula. You'll arrive at 12 primes in all, including the original 6 of 9, bringing the 2/3 result down to 12/33 or about 36%. In all, for both sides of the 10 line then, you'll find 23/65 primes, or about 35%.
3. Investigate further. Repeat the process for other lines which show promise of containing primes. One such line is the series 3, 5, 19, 41, 71, 109, 155, 209, 271 (all are prime except 155 and 209 for a ratio of 7/9 or about 78%) as runs from the center to the upper left corner. You'll notice that, except for the beginning 3, it is one unit away from the diagonal of 2*3, 4*5, 6*7, etc. -- all that is needed is to add a decrement of -1 while adding 2 to each integer multiple each time for the extension.

   1. Try extending the series to 32 members; you'll find 20 primes in all, rendering a ratio of 20/32 = about 63%.
   2. Then extend the lower right part of this diagonal to 32 members. Its series is 3, 13, 31, 57, 91, 133, etc. and its formula is 1*2+1, 3*4+1, etc. So by adding 2 to each of the integer multiples each time and carrying the 1 and Formula, you will quickly be able to extend the series to 32 members. You'll find 11 primes and 11/32 = 34%. For the entire diagonal then, you should obtain (20+11)/(32+32) = 48%.
4. Investigate the last line. This is the series running 5, 17, 37, 65, 101, 145, 197, 257 and 7, 22, 47, 79, 119, 167, 223, 287 -- both returning a figure of 75% primes for 8 numbers invested each time. Extend both of these and summarizing, you should get (13+19)/(32+32) = about 50%.

   1. Therefore, these diagonals and rows proved particularly rich in primes, even as primes became more rare. If they are not "Predictors" of primes per se, they are at least good indicators.

Further Checking

1. Check against the Known Primes List. So far this article has not mentioned the rather tedious task involved hereby of having to check each extended number against a Known Primes List. There is also nothing inherently valuable in the process as to establishing whether a number is prime or not, except that it might shortcut that process somewhat. It could, by a thorough approach, investigate all integers but that would be the long way around. To do so, you merely need to build a generating formula for each column and/or row and keep extending it, while doing so in a methodical sweep that takes in every integer. Else, approach the matter as a series of (split?) diagonals. The initial author of this article has not tried this yet, so it is not confirmed whether outlying diagonals need be split at the axis or not as to their formulas but it may well be so, per quadrant entertaining the diagonal.
2. It may be that by some keen usage of Continued Fractions, in combination with the Spirallic Dataset, for primes vs. Non-Primes, a key discovery might be made, since almost every number (AEN) is susceptible to cogent representation by continued fractions. it's worth investigating.

Concluding

1. Expect a big decrease. All in all, then, you might expect to experience about a 40% loss in the number of primes found as you move from the range of 400 to 4000, approximately, when following the most probable lines of inquiry indicated by a spirallic dataset. The primes page here shows that there exist 586 primes at 4271, or a ratio of 13.72%, versus 76 at 384 or 19.79%. Of the 586 primes at 4271, the initial dataset and thrust along probable lines, reduced for double-counting, captured 131 of them, or 22.4% (the initial 66 being 20% of 384 and 131-66= 65 of the remaining 586-66-520 so 66/520 = 12.7% were captured by the 6 probes along 3 split lines). You might easily improve upon these results as there are more than several strong columns, rows and diagonals indicated which were not pursued in this article. That said, it was not all that long ago that the world's largest prime was found, containing over 17 million digits!

Helpful Guidance

1. See the article How to Create a Spirallic Numbers Dataset for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.

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