[^^mac] [Back to the MATHS page] (Thanks and 42 flips of the towel to Carl Sagan see note below as to why)... There are certain questions concerning the randomness of primes (or at least their apparant distributions)... The density of prime-twins, Also, the density of primes that end in say "23" or "51" or "97" For each block (eg, 1-10k, 10k-20k, 20k-30k etc is there a "more probably last digits set". And as we go further out, we could look at the distribution of the last 3-digits of a prime. These again lead back to the randomness of prime distributions. One almost wants to ask questions of a "Monte Carlo" nature about the distribution of primes, and the digits "patterns" in pi. Is there a prime expression for pi (other than the classic one), and why is it a "reciprocal" relationship. The division function "recapitulates" itself in the natures of divisibility (prime-ness) and then again in terms of the "pi-ness" of things? All very odd. And of course the "distribution" of ending-digits is based on the fact that we are using BASE-10 -- that's the reason why there won't ever be any last digits of primes like: 05 15 25 35 45 65 75 85 95 <-- these occur because the base has "5" as a factor 10 20 30 40 50 60 70 80 90 <-- these occur because of the base This begs the question of whether there is a "natural" base that should be used. That is: Is there something about numbers that shows that there is a "prefered" base? In theory, it should turn out that the answer is no of course. Consider expressing numbers in BASE 3, then by the above list and a "similarity" argument, we should expect that we would not ever see any primes ending (in base 3) with: 10 20 The primes in base 3 are: 2 3 5 7 11 13 17 19 23 29 31 02 10 12 21 102 111 122 201 212 1002 1011 ^^^^^^^ Note the rare occurence of two consecutive primes that are both "even" (ie, end in 2 -- but "even-ness" doesn't mean in base 3 what it does in base 10!!! Thus, while it looks like "212" (base 3 for "23") is a multiple of 2 ("2" in base 10), the division reveals this (is this complicated enough??? ;) Performing the division in BASE 3.... 1 0 2 +------------ 2 | 2 1 2 - 2 --- 0 1 2 - 1 1 ------ 1 <-- remainder of 1, there-fore 212 (base 3) is NOT a multiple of 2. The pattern is clear: No primes greater than "3" (10_{3}) will ever end with a zero. (Otherwise they'd be a factor of 3, just like n*10 ends with a zero in base 10). In general, the primes tend to alternate between ending in 1 and then 2. Thus, we seek (in addition to twin-primes), occurences of consecutive primes that end in the same digit. It "seems" like the patterns in one base may (or may not as Broomfondle might well say), mirror the patterns in another base. Of course, all of these questions are just as "mysterious" as the yearnings to find the "utlimate" volume in the Library of Babel. Is there even any point in asking such questions? Hmmmm... btw: I can not take complete credit for this idea of their being "patterns" in the various bases. In his novel, "Contact", Carl Sagan discusses that the universe was "constructed" and as such the number pi (out about a billion places or so), when displayed in base 13, displays a very NON-random pattern: A string of ones and zeroes that end up forming a picture of sorts. (Of course, it's left to the imagination of the reader as to whether this means that there IS a god (who designed pi and hence the universe -- which came first pi or the universe?), or that it's just a "coincidence" -- that is, if you keep trying combinations of BASE's and look long enough into the value of pi then sooner or later you'd find SOME pattern). So, maybe the search for pi (primes?) are the search for the divine, for some "deeper meaning" in the structure of the universe -- or at least in the "mathematical universe". So, instead of asking ourselves how many angels can dance on the head of a pin (the answer of course being "as many as want to"), we now ask ourselves, how many digits of pi does it take to make a rose? or a DNA molecule (even with our without a towel).... Back to the MATHS page Back to the MAC home page