/ B(n) = (10^(n+3) + 666)*10^(n+1) + 1 /
1030 + 666 × 1014 + 1
Belphegor’s prime is the palindromic prime number which reads the same both backwards and forwards and is only divisible by itself and one, represented by the Greek letter Pi π upside down. The name Belphegor (Ba’al peor, בעל פעור) refers to one of the Seven Princes of Hell, the Lord of the Gap who was charged with helping people make ingenious inventions and discoveries. It’s name is coined by author Clifford A. Pickover but actually first discovered by prime number hunter Harvey Dubner, who determined that it was part of a sequence. Why Pickover chose Belphegor exactly is unclear, but he warns people not to stare at the number too long and to look away and take a deep breath after a few seconds. He also links the number to a symbol from the mysterious Voynich Manuscript, represented in the same manuscript by an inverted pi symbol, too.
The set of primes could be reached by altering the number of zeroes on either side of the 666 to appropriate amounts. This is to say that the number 16,661 with no zeroes at all is also a prime number. Then the next time a similar palindromic number becomes prime is when you have 13 zeros (overall decimal number length of 31-which is 13 backwards), then again when you have 42 zeros, etc. Harvey Dubner determined that the first 7 numbers of this type have subscripts: 0, 13, 42, 506, 608, 2472 and 2623.
In the short scale, this number would be named:
one nonillion sixty-six quadrillion six hundred trillion one
But in the long scale, this number’s name would be:
one quintillion sixty-six billiard six hundred billion one
The number itself contains superstitious elements that have given it its name: the number 666 at the heart of Belphegor’s Prime is widely associated as being the Number of the Beast, used in symbolism to represent one of the creatures in the Apocalypse or, more commonly, the Devil. This number is surrounded on either side by thirteen zeroes and is 31 digits in length (thirteen reversed), with thirteen itself long regarded superstitiously as an unlucky number in Western culture. In wider demonology, Belphegor is a demon who seduces people by suggesting to them ingenious inventions that will make them rich. 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. Belphegor also figures in John Milton’s ”Paradise Lost” and in Victor Hugo’s ”Toilers of the Sea”.
”Belfagor arcidiavolo” by the Italian diplomat Niccolò Machiavelli was first published in 1549 and regales how the demon comes to earth to find a mate. This story became the initial source material for Ottorino Respighi’s opera Belfagor which premiered at La Scala in Milan in 1923.
Clifford Pickover, in his book ”Keys To Infinity”, defines numbers whose product is a permutation of their digits. For example, 27 and 81 are vampire numbers because: 27 x 81 = 2,187