One of my interests in pi, and this is all purely recreational, is locating unique integers/relationships, looping numbers ( " orbits " ) etc. In pi, certain integers are located with a position number that when multiplied, by single integers, will return the number itself as a product. For example, the square of 12: { ...77235014144197356854... }: 144 at the 1638th position 1*6*3*8 = 144 My question is this: Is there a simple way to determine which integers have this relationship with their position numbers, in pi ? I already know I can disregard any position number with 0 as a digit, as it will give 0 as a product. As a string grows larger, say instead of a 3 digit integer like 144, I search for an 8 digit integer, like 14444444, the frequency with which the string occurs, becomes less and less frequent, ..so is it reasonable to assume that the probability of finding that particular relationship between a string and the multiplicative product of it's position number, becomes less and less probable as the string grows larger ? How would this apply for pi in other bases, if at all ? Thanks, Isaac
There is no way at all to predict any special sequences and relationships in the digits of pi other than calculating the digit sequence and looking. From what you've written, that seems reasonable ... if N is the position of a particular sequence of 3 numbers, then the bigger N is, the more digits it has to multiply together, the more change that the product of the digits will be bigger than three digits. There's nothing special about the digits to pi - this is a property of any large sequence of digits in any base. Note: it is not clear what is meant by this the position number. i.e. is it; (a) place 1: 314, place 2: 141, place 3: 415 ... etc (b) place 1: 314, place 2: 159, place 3: 265 ... etc (c) place 1: 141, place 2:... i.e. as above but only counting decimals. (d) place 1: 003, place 2: 031, place 3: 314, place 4: 141 ... ... but, basically, this is all just numerology I'm afraid. Don't be fooled: people have looked for sequences and retrofitted the significance.
By " position number ", I mean the digit place which locates a particular integer, like the example I gave. There were three digits in the integer 144, they were at the 1638th ( 1 ), 1639th ( 4 ), and 1640th ( 4 ) digit positions, after the decimal. I'm not too sure what you mean by " numerology " here though, I usually try to keep my forays into recreational mathematics and " numerology " in a separate basket. I'm more interested in things that could be considered a mathematical " hapax legomenon ", that's all.
So when you said "144 is at the 1638th position" What you meant was that the "1" in "144" is at the 1638th position. That would be option (a) in my list ;) hapax legomenon? You mean you are looking for a sequence of numbers in the digits of pi which occurs only once? Good luck. It's numerology - attaching an undue significance to coincidental relationships between random numbers.
Oy vey Well, Simon, that's where you and I differ in our definitions of " numerology ", apparently. I tend not to lump completely unrelated topics together with generalizations. Having had to wade through an inordinate amount of fluff in the process of writing my book, I am quite clear of the differences in the subjects, whereas people who toss out that term usually do not even know the fundamental differences between something like ELS ( actual numerology ) and chiasmus ( a syntactical structure ) If you wanted to discuss the history of encryption and ciphers, which involves the study of the evolution of alphabets, the proper term is actually " semiosis ",... not " numerology ". Since I've already stated, twice now, I think, that I am not interested in " mystical " meanings behind numbers, I would hope that I am being clear. Like always, I appreciate an opinion though. It makes for a great discussion in itself, especially if you have interest in the work of people like Eco Umberto or Bertrand Russell. ---------------- In regards to my interest in pi, like I said already, it's purely recreational. No different than what you'd find in the " trivia " section of a website like Pi search page Thanks again, Isaac
There are "spigot" formulas, such as one to calculate the nth hexadecimal digit of the fractional part of pi. Wiki article: http://en.wikipedia.org/wiki/Pi#Spigot_algorithms