Thursday, October 30, 2014

Halloween Appropriate Mathematics

Since tomorrow is Halloween, I thought we would address some Halloween Appropriate Mathematics.

Beastly Primes
The Prime Glossary defines two kinds of Beastly primes.  The first type are palindromic primes that have a 666 at the center, which is surrounded by zeros, with either ones or sevens on the end.  The second is a prime number that starts with 666, which is followed by a string of zeros, with either a one or a seven on the end.
The Online Encyclopedia of Integers sequences also describes a set of palindromic primes that have “666” in the middle of the prime (but they do not call it a Beastly prime.  My favorite is 3166613 because it includes the number of the beast, thirteen, and a backwards thirteen.  Spooky!
And they also describe Beastly prime cousins that occur when p is a prime number, and p + 666, and p – 666 are both prime also.  I am not sure if they are referring to all three primes being prime cousins or just the two (p + 666, and p – 666) formed from the original prime p.
Belphegor’s Prime is the most famous Beastly Prime:  1,000,000,000,000,066,600,000,000,000,001.  It consists of a one, 13 zeros, “666”, 13 zeros and another one.  If the 13 ones, and the 666 is not spooky enough, the name Belphegor should be.  Belphegor is one of the seven princes of hell.(representing vanity and sloth, and responsible for helping people make great discoveries or inventions).  This number is also a Palindromic prime and a Naughty prime.  And it has 31 digits (which is 13 backwards).
Belphegor - not your math teacher from last year.

What is interesting is that this number remains prime if you substitute the thirteen zeros on either side of the “666” with 42, 506, 608, 2472, 2623, or 28,291 zeros on each side of the “666”.

Vampire Numbers
Vampire numbers have an even number of digits, and the digits can be rearranged to form two fangs (or factors) whose product is the original number.  For example 1260 is a Vampire number because the digits can be rearranged into 21 and 60, and 21 * 60 = 1260.  OEIS also has a definition of Vampire numbers which can include number with an odd number of digit, fangs (factors) that are not equal sized (have a different number of digits), and might have more than two fangs.  I think a vampire with 3 fangs with a big one on the right just looks funny. 

Witch of Agnesi
Maria Gaetana Agnesi
Maria Gaetana Agnesi was an Italian mathematician and philosopher, who lived in the 18th century.  She was an honorary member of the faculty at the University of Bologna (the university did not teach bologna, it was located in Bologna, Italy).  She is credited with writing the first book that covered both differential and integral calculus.
The "Witch of Agnesi" is the cure going through points M and P.
In her book “Instituzioni analitiche” she discussed a curve earlier studied and constructed by Pierre de Fermat and Guido Grandi.  Grandi called the curve “versoria” (a nautical term meaning “sheet”), but somehow the term was mis-copied or mis-translated into “versiera” which means “she-devil” or “witch”.  So the curve has become known as the "Witch of Agnesi".  It’s not a very scary curve, but if you dressed up as the “Witch of Agnesi” very few people would be able to figure out what you were.
You might want to go see this short video:



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