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.

Dear Sabio/Son of Sabio,

I am a mathematician, and I can answer your questions.

1) First question : what is it with the 25 and 35 phenomena?

Well, the answer is very simple. When you make the digital sum of a number n, you get a new number (that i will denote by ds(n)), which is much smaller (except if you start with a single digit number), and which has an interesting property : the remainder of the division of n by 9 is the same as the remainder of the division of ds(n) by 9. (we say that n and ds(n) are congruent modulo 9)

So, if you test every prime number less than 100000, you will obtain new numbers which are less than 9+9+9+9+9 = 45 (exercise : undertand why this is true). Now, let’s have a look at odd numbers between 2 and 45 : 2,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45.

If n is prime, then n can’t be divided by 3 (except if n=3), and ds(n) can’t be divided by 3 (exercise : explain why). Therefore, ds(n) can’t be equal to 9,15,21,27,33,39 or 45. The only numbers left are 3,5,7,9,11,13,17,19,23,25,29,31,35,37,41,43. These numbers are all prime except 25 and 35.

So, basically, the answer to your question is that 25 and 35 are the only non-prime numbers in the list of odd numbers between 2 and 45 which are not divisible by 3. It comes from the fact that there are a lot of prime numbers in the beginning of the natural numbers.

However, the next non-prime number not divisible by 3 is 49. Then it is 55, 65,77, 85, … It seems very likely that you will find prime numbers with those values as digital sum. However, you will need to go further. To have a digital sum of 49, you will either lots of digits, or lots of 7/8/9 in your digits.

Bonus information : being prime number is a property of a number. However, the digital sum operation depends on the choice of a basis to write your numbers in. In your case, you chose base 10 and 3 is the magical number that appeared in this argument. But if you were to write numbers in other bases, for example base 16 (hexadecimal), you will get different “magical numbers”. In base 10, the magical number was 3 because 10-1 = 3*3. In base 16, the magical numbers will be 3 and 5, because 16-1 = 3*5. In base 50, the magical number will be 7 because 50 – 1 = 7*7.

2) Second question : links between the digitalroot function and the number 3.

Here again, the answer is simple. Basically, the digital root function is the remainder of the division by 9, so if it is 3, 6 or 0, it means that the initial number was divisible by 3, and so, couldn’t be prime in the first place.

Here again, the number 3 appears because human ppl have decided to use base 10 to write numbers. As above, you could choose different basis and you would have different magic numbers. For example, as above, if an alien race had only 6 fingers and decided to use base 6, then 5 would be their number and they might believe in some divinity with 5 theological aspects (how do you say trinity, but with 5?)

If you want more information, just email me.

@ GDWow, thank you for the clear answer. Well done. So it appears that all I have revealed is the unique property of DigitalSumming and nothing about reality. 🙂 Color me startled. And my easily amazement still helps Michael Shermer make his point!

Seriously, that was fun and embarrassingly obvious. I really did not expect to have found an amazing discovery but still found it fun.

Your “Bonus Information” was fun. I have thought about this before and realized that working is different bases is a good way to see through local tricks vs. wide phenomena. Likewise, on my blog, I feel that understanding several religions, several languages and several medical systems helps one to see deeper patterns than merely trying to understand the world having only lived in a small pond.

Concerning my question about the number “3”. You have shown me that THREE (the Trinity) is only a local god — indeed, my suspicion. 🙂

Thanks GD — that was fantastic. Much appreciated !!

May we ask where you teach or work as a mathematician and if you have a web site? And how did you find this post?

I like your way of thinking about local trick/wide phenomena. It is an appropriate way to see it. And seeing about the trinity as a local god seems interesting. Good food for the brain. 🙂

PS : I prefer keeping my “digital footprint” as small as possible. I’ll answer your question privately.

I like G.D.’s answer. I was going to respond with a much less detailed answer: take a look at divisibility rules, which can be mundanely proven, and then consider that sums of digits of primes higher that 1,000,000 will of course have more digits, so your observation is unsurprising given that sums of digits are useful in divisibility tests, and you were dealing with two-digit sums.

Now, here is a much more interesting question. Does the entire text of the King James version 1611 version appear somewhere in the digits of Pi? It turns out, there is a good possibility that it does. ZOMFG! What does it mean?!?

@GDThanks again

@ JS AllenIndeed, there is a smaller chance that the entire Bhagavad Gita is inside of Pi but a higher change that the holy Quran can be found encoded in Pi. All are possible. Since they all possible exist, how can we choose which to be amazed by. BTW, I am sure also that Marquis de Sade’s “La Philosophie dans le boudoir” is also to be found in Pi.

Actually, it’s probably all or nothing. If the KJV is in there, then the Baghavad Gita, Quran, and all of the collected works of Marquis de Sade are in there too. And all of the future contents of your blog 🙂

All it depends on is knowing that Pi is completely normal (we think this is true, just haven’t proven it). If so, then every possible finite string appears in there eventually.

wow, thanks, G.D.

I felt compelled to work out the answer – you saved me a lot of time!

(Sabio, don’t you know that readers hate an unanswered mystery? That’s why all good books explain the plot at the end.)

Clearly, there are 60 (25+35) other gods out there. This is why, when you multiply the number of fingers god(s) created you with by the number if apostles, you get sixty. We have found the truth!

Holy crap. That was the most rational and well-written explication of anything mathemolatical I’ve ever had the pleasure of reading. Well done, G.D.

And Sabio, my friend: back to the drawing board, I s’pose. You’ll find yourself a Goodly Fere one of these days.

@crl – 🙂

More seriously, does anyone have a good source to recommend on the significance of certain “special” numbers in ancient Hebrew and Babylonian? I’ve read a few books that explain how things can go horribly wrong with numerology in recent times, examples of ancient Greek numerology craziness, etc. But I’m looking more for a simple “Here is all of the connotations that would come to mind for the number 12, if used by a well-educated Babylonian priest”.

@ JS Allen:Pi Mysticism — we have much in common!

Sorry, can’t help you with your desire to dig into Hebrew numerology — are you planning to set up a Calvinist Baptist radio station to tell your numerical predicitions? 🙂

Trying to supplement your daytime job?

@ Boz:Unsolved mysteries is like catnip for atheists

@ Zachary:I have not despaired and am still searching for little winged creatures.

@ crl:Indeed! Praise 60 !

@Sabio – Haha, I hope it never comes to that! Just trying to get in the frame of mind of the guys who wrote those books. To them, numbers had some significance that they don’t have to people today.

@JS Allen

If you’re interested in learning more about Hebrew numerology, you might look into Qaballah. It deals heavily with numerology if you can find unbastardized, non-new-age sources to work with.

@ JS: 12 days of Christmas & 12 leads on an EKG machine to see if you are having a heart attack when the credit card bill comes in.

Connection? The bill for the presents comes to the house on the 12th day after Christmas.

The wisdom of the Ancients lives on!

Cool – I like some Math – but this was a totally interesting read!

I’d like to add, in case anyone else didn’t, that you cannot have a prime, who’s numerology equates to 3 or 9, except for 3 itself. Any number that equates to 3 is wholly divisible by 3 and any number that equates to 9 is wholly divisible by 9.

As to your 25 & 35, they are both numbers of the 23 Trinity:

25 = 5^2 = 2^5 = 32 = 3 + 2 = 2 + 3 = 23

35 = 3 + 5 = 8 = 2^3 = 2 + 3 = 23

I hope that makes you think a little more about what you found. 😉

Ribbit 🙂

Ps: I wrote an interesting Numbers Puzzle that you should enjoy pondering over. Email me and I will send it to you.