Sunday, 24 February 2013

A Devilish Prime, The Devil is in the Details

After years of looking into the origins and evolution of mathematical terms, ideas, and recreations, it has worked its way into my thinking process. So when I recently saw a post on a page by Clifford Pickover about Belphegor's (Sometimes Belphagor) Prime, I was intrigued. I love mathematical oddities, I had never heard of Belphagor (or his prime), and a big picture (above) on the page also caught my interest.
So First, what it is: It seems that Belphagor's Primes is the number 1000000000000066600000000000001, and as advertised, it's a prime. This beautiful palindromic prime has a 1 at each end, with 666, the number of the beast in the middle, and thirteen zeros on each side separating the 666 from the units. The symbol even had its own symbol, a sort of inverted π. The symbol charles berry solitare pegitself comes from something called the Voynich Manuscript, which I had also never heard of, and which has its own strange history.  All in all, so much to explore. 

The number itself is interesting in that it is one of a sequence of primes.  It seems that 16661 is prime also, and 1xxx666xxx1 is prime if the xxx is replaced by an appropriate number of zeros. Harvey Dubner determined that the first 7 numbers of this type have subscripts 0, 13, 42, 506, 608, 2472, and 2623 [see J. Rec. Math, 26(4)].
16661 is an interesting prime itself it is one of a special case of primes for which the sum of its decimal digits is the same as the sum of its prime index. 16661 is such a number, since it is the 1928th prime, and 1 + 6 + 6 + 6 + 1 = 1 + 9 + 2 + 8 = 20

Dubner is himself a little known (certainly relative to his merits) individual. A a semi-retired engineer living in New Jersey, he is noted for his contributions to finding large prime numbers. In 1984, he and his son Robert collaborated in developing the 'Dubner cruncher', a board which used a commercial finite impulse response filter chip to speed up dramatically the multiplication of medium-sized multi-precision numbers, to levels competitive with supercomputers of the time, though nowadays his focus has changed to efficient implementation of FFT-based algorithms on personal computers.
He has found many large prime numbers of special forms: repunits, prime Fibonacci and Lucas numbers, twin primes, Sophie Germain primes, and primes in arithmetic progression. In 1993 he was responsible for more than half the known primes of more than two thousand digits.
In addition, he is credited with the invention of the first blackjack point count (The High Low Count) which is used by most blackjack card counters today.

I haven't yet been able to find a copy of the journal above, so I don't know if he was the first to find Belphagor's prime, or if he just extended a known streak.

So some questions come to mind: who first found that the number was prime, who/what was Belphagor, who named the Prime Belphagor and when, What's up with the upside down π symbol, and what is Voynich's Manuscript and what does it have to do with Primes or Belphagor or Pi?

Some of these questions still remain unanswered, so consider this as much a request for information as a presentation of such.
Belphegor, it seems, was/is a minion of the Devil. Wiki says he "is a demon, and one of the seven princes of Hell, who helps people make discoveries. He seduces people by suggesting to them ingenious inventions that will make them rich. According to some 16th-century demonologists, his power is stronger in April. Bishop and witch-hunter Peter Binsfeld believed that Belphegor tempts by means of laziness. Also, according to Peter Binsfeld's Binsfeld's Classification of Demons, Belphegor is the chief demon of the deadly sin known as Sloth in Christian tradition." So how did I guy as lazy as I am not know about this guy??? Just to lazy to look him up I guess.
And by the way, the picture above.... It seems that is NOT him. The image above is from ST. WOLFGANG AND THE DEVIL by Michael Pacher and is part of the magnificent 15th-century Altar of the Church Fathers, now in the Alte Pinakothek in Munich. The Wikipedia image of Belphegor shows him apparently sitting on a toilet. It seems "According to De Plancy's Dictionnaire Infernal, he was Hell's ambassador to France." It is from De Plancy's book that this image of Belphegor is found. Belphegor is sometimes associated with Ba‘al Pe‘or a God of the Moabite people in Numbers 25 in the old testament of the Bible.

Ok, so next I tried to track down the mysterious upside down π glyph that was used as the numbers symbol.  It appears, the post says, in the Voynich Manuscript.  This turns out to be an intriguing mystery of its own.
Wikipedia again, tells me that it is called "the world's most mysterious manuscript".
,is a work which dates to the early 15th century (1404–1438), possibly from northern Italy. It is named after the book dealer Wilfrid Voynich, who purchased it in 1912. Some pages are missing, but the current version comprises about 240 vellum pages, most with illustrations. Much of the manuscript resembles herbal manuscripts of the 1500s, seeming to present illustrations and information about plants and their possible uses for medical purposes. However, most of the plants do not match known species, and the manuscript's script and language remain unknown and unreadable. Possibly some form of encrypted ciphertext, the Voynich manuscript has been studied by many professional and amateur cryptographers, including American and British codebreakers from both World War I and World War II. As yet, it has defied all decipherment attempts, becoming a famous case of historical cryptology. The mystery surrounding it has excited the popular imagination, making the manuscript a subject of both fanciful theories and novels. None of the many speculative solutions proposed over the last hundred years has yet been independently verified.
Wow, cool, but with seemingly nothing to do with primes, demons, or math other than the cryptographic problem ???  At least we have a background date.  Whoever and whenever the symbol was attached to the prime, it was after 1912 when the Voynich document was purchased. 

In fact there seem to be no occurrences of "Belphegor's prime" in a Google Book search, and none of the hits on a general web search dated before 2012, so it may well be that the name and use of the symbol are creations of someone, perhaps Clifford Pickover, that has occurred very recently.

So I ended up with more questions than answers, much like my ill-fated search for Gauss' pipe.  But I did come across with a page with several interesting relationships involving the infamous 666 by a gentleman named Mike Keith. I have included a few I found interesting below. If these fail to satisfy, he has a plethora of other "Beastly" offerings here. :


666 is equal to the sum of its digits plus the sum of the cubes of its digits:
666 = 6 + 6 + 6 + 6³ + 6³ + 6³.
There are only 6 positive integers with this property.


The sum of the squares of the first 7 primes is 666:
666 = 2² + 3² + 5² + 7² + 11² + 13² + 17²

The triplet (216, 630, 666) is a Pythagorean triplet. This fact can be rewritten in the following nice form:
(6·6·6)² + (666 - 6·6)² = 666²

A well-known remarkably good approximation to pi is 355/113 = 3.1415929... If one part of this fraction is reversed and added to the other part, we get

553 + 113 = 666.

[from Martin Gardner's "Dr. Matrix" columns] The Dewey Decimal System classification number for "Numerology" is 133.335. If you reverse this and add, you get

133.335 + 533.331 = 666.666

And long after I first was inspired to write all this by a post from Clifford Pickover, I saw another reference to 666 in one of his tweets:  "666 hides among 0s in Pi! The string 006660000 occurs in Pi at position 58,488,501 counting from the first digit after the decimal point."


A short time after I wrote this blog, I got a nice note from David Brooks with some information about this number that I was unaware of (0k, no surprise, lots of stuff I haven't figured out) which I share below:

"Some trivia about this prime that you may (or may not) be interested in. First, it is a "naughty prime" - it is composed mostly of "naughts" or zeros. Second, it is a "Repulican prime" - the right half (ignoring the middle digit) is a prime number, but the left half is not prime."

The OEIS sequences for both numbers are here :
Naughty Primes: http://oeis.org/A164968 10007, 10009, 40009, 70001....

And Republican Primes: http://oeis.org/A125524/internal 13,17,43,47,67,83,97,103,107,...

and for political equity, I should point out that there are Democratic primes as well, defined as you would expect from the definition of Republican primes above, http://oeis.org/A125523/internal
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