In Skepticblog, Michael Shermer just posted a fun article about his recent run-in with numerology. He tells of an uncomfortable fake interview he had with a proud self-proclaimed Muslim heretic from Kazakhstan who staged the interview with Michael in order to push his Islamic book about the mystical implications of the number 19. At the end of his article Shermer challenges readers to “employ their own patternicity skills at finding meaningful patterns in both meaningful and meaningless noise with such numbers and numerical relationships….” In Shermer’s terms, “paternity” is the hyper-over-kill ability to see connection and meaning where it doesn’t exist. Coincidentally (hmmmm?), I just recently discovered an miraculous numerical pattern which I am entering in Michael’s challenge. See if you agree that it is miraculous. Click more to read my inspiring story:
This summer I am teaching my 11-year-old son how to do computer programming. Our first projects were to build a simple calculator that asked the user to supply their weight and their favorite planet and then the program calculated the user’s weight on that planet. Next it was time to teach him about the power of “control flow” (AKA: “looping”). For this we built an algorithm that listed prime numbers. My son was fascinated by how fast the computer could find primes — he had done some by hand and seen how tediously slow the task was to sieve-out primes from the whole numbers. He was also dazzled to see that it took our computer 2 hours to generate all the primes up to one million.
We decided to use the computer to further explore number properties. We wrote a program which took any number and summed the digits in the number — “digit summing”. For instance, the DigitSum of 294 = 2+9+4 = 15. Next we looked at how to link our two algorithms — our prime number generator and our digit summing algorithm.
We had this new combined algorithm program generated a list of the DigitSums of the primes up to 500, and to our surprise, an unusual pattern jumped out! As we looked at the list, we noticed that the odd sums in this list were all primes — of course the even sums were not prime.
We were tempted to settle in self-satisfied amazement and stop there, but with scientific doggedness, we decided to look further. We ran the output of that series through our prime number filter algorithm and ran the series through 1000. Damn! 997, the last prime in that range, had an sum that was not prime – “25”. We were disappointed — our great discovery had been ruined. Dispair fell upon us. But we decided to go further. We ran our algorithm up to 100,000 and low and behold, the only prime sums-of-primes were either 25 or 35 ! OK, we were excited again.
So, why are 25 and 35 so unique? What is that about? I searched the web a bit but have not yet found others who discovered the same. I re-scanned my notes in the margin of the book “Prime Obsession” by John Derbyshire and re-read a few chapters, but found nothing. But I am not a mathematician and not sure where else to look. Have we discovered a deep secret of creation?
Enamored by the DigitSum function, I decided to explore the DigitalRoot function which takes DigitSums a step further. DigitalRoot is the result of using DigitSums on itself until the result of any given number is a single digit. For instance: 46239 => 4+6+2+3+9 = 24 => 2+4 = 6. Using DigitalRoot, I decided to see if the primes had another weird pattern like the 25 and 35. Bang! Sure enough, the DigitalRoot of primes failed to generate 6s and 9s and only rare 3s whereas the other digits were in equal distribution. Wow!
But being a good skeptic, I wondered if this was just a property of the DigitalSum function and not a property of the primes themselves. So next I generated the DigitalRoot series of whole numbers up to 100,000 and plotted their frequency. Boring ! It doesn’t take much to realize that the DigitRoot would be equally distributed between 1 and 9 for all whole numbers. So I examined the series of even numbers and odd numbers — same even distribution. So far, it seemed the primes were unique. But then I discovered a surprise when I generated a series where I counted by three or any multiple of 3 — holes appeared in the distribution of the DigitalRoots. So it seems that it was not primes that were unique in this property but instead it was the DigitalRoot function and the number 3. Then it hit me: The Trinity! Here was the Holy Trinity staring me in the face. God had again appeared before this unrepentant Skeptic.
OK, I understand that Jehovah of the holy Trinity should display himself in mathematics, but my heretical mind took over again: What is it with the 25 and 35 phenomena I discovered in Prime Digit Sums? Is there another god to be found out there? :-)
I am pretty sure this is what Michael Shermer was looking for in his challenge — to show how the superstitious mind unabashedly employs patternicity.