I wrote a program to compute successive approximations of Pi using the following Euler product:
π/4 = (3/4)*(5/4)*(7/8)*(11/12)*(13/12)...
in which the numerators are the odd prime numbers, and the denominators are the closest multiples of four (picture here).
I only printed out approximations which were better than all previous approximations. For example, if we define term b(n) as a(1)*...*a(n), we get b(1)=3, b(2)=3.75, b(3)=3.28125, b(4)=3.0078125, b(5)=3.2584635416, b(6)=3.462117513020833, b(7)=3.289011637369791, b(8)=3.1519694858127165. Of these, only terms 1, 3, 4, 5, and 8 lead to better approximations, whereas terms 2, 6, and 7 do not.
Here are the results for b(1) through b(5e9). Each line lists n (the number of terms), the n-th odd prime (i.e. the numerator), and b(n) = n-th approximation of Pi.
My main question:
Why are the majority of terms spaced an even number of primes apart? You can see this in the long runs of odd terms (including the last 38) as well as runs of even terms. Out of 71 terms listed below, only 12 are an odd number of primes apart (and half of those are consecutive primes):
n nth prime b(n) -------- --------- -------------------- 1 3 3 3 7 3.28125 4 11 3.0078125 5 13 3.258463541666666667 8 23 3.151969485812717015 47 223 3.142881020579148847 49 229 3.142820561956869316 95 503 3.142133362655805143 247 1571 3.141912463610110751 251 1601 3.141870207419266742 253 1609 3.141868992307710322 742 5651 3.14162426815169492 4268 40771 3.141615756075984207 4270 40801 3.141615731534329301 4288 40961 3.141615677490338508 11445 121607 3.141598027134934751 30123 351863 3.141590250585033286 30701 359207 3.141592343379190447 30703 359231 3.141592343939149987 62592 781631 3.141592401246138009 62690 783019 3.141592434797031353 62992 787091 3.141592555331437549 3535871 59530267 3.141592744321108717 3535872 59530291 3.141592691548097889 3664203 61831547 3.141592690253390274 3664204 61831579 3.141592639444520594 3664214 61831747 3.14159263944470384 3664220 61831927 3.141592639444941319 3665670 61857923 3.141592639728499004 3665696 61858267 3.141592639729109027 3665842 61860691 3.14159263973972529 3665854 61860947 3.141592639739733499 3665866 61861159 3.141592639739846788 3708907 62634323 3.141592650836013723 3708909 62634331 3.141592650836016126 3708913 62634379 3.141592650836065776 3708929 62634631 3.141592650836687188 3708931 62634643 3.14159265083668959 3708935 62634763 3.141592650836755255 3708957 62635171 3.14159265083739669 3708983 62635603 3.141592650838468924 3708985 62635663 3.141592650838490544 3709017 62636359 3.141592650840072844 3709025 62636503 3.141592650840184947 3709031 62636603 3.14159265084026262 3709335 62641987 3.141592650886839043 3709529 62645567 3.141592650902843015 3788299 64058429 3.141592655663886643 3788307 64058573 3.141592655663770271 3788315 64058693 3.141592655663641654 3788349 64059221 3.141592655662596635 3788357 64059329 3.141592655662480267 26172875 496334123 3.141592653356849149 26172877 496334171 3.141592653356849699 26175227 496381447 3.141592653365838828 26175231 496381511 3.141592653365838903 26175239 496381651 3.141592653365841606 26175359 496384127 3.141592653365985622 26175361 496384219 3.141592653365985865 26175383 496384723 3.141592653365987204 26175619 496389431 3.141592653366111398 26176783 496412687 3.141592653369742264 26176787 496412771 3.141592653369743412 26176789 496412779 3.141592653369743501 26179059 496458439 3.141592653384679576 26179063 496458499 3.141592653384680369 26179191 496461071 3.141592653384854573 26180229 496481387 3.141592653390104058 26180235 496481471 3.141592653390104479 26180239 496481639 3.141592653390106083 26180241 496481663 3.141592653390106375
The 12 terms with odd spacing - often '1' - mentioned above: (3,4)=>(7,11), (4,5)=>(11,13), (3535871,3535872)=>(59530267,59530291), (3664203,3664204)=>(61831547,61831579)
Other questions:
- What is the distribution of terms in this sequence? Does this sequence have characteristics that differ from than that of the Euler product containing all terms? Why are there such large gaps between successively better approximations, followed by clumps of improvements? For example, after term 62,992, there are 3,472,879 intervening terms before the next improvement - after term 3,788,357, a whopping 22,384,518 terms before the next improvement - and at least 10 billion terms (and counting) before whichever one comes next. Although the series converges, at various scales it resembles a random walk - is there a better or more precise way to characterize it?
Thank you for your time -
I'm afraid the answer to your main question is a bit more mundane than one might have hoped.
There are up and down steps, and for large numbers they're all very nearly the same size. The net change due to $n$ up steps and $n$ down steps is far less than the net change from any unbalanced set of steps. To an increasingly good approximation, you can think of the distance to $\pi/4$ as a random walk with the fractional part drifting much more slowly than the integer part – relative to the current step size, but that step size changes ever more slowly.
The improvements occur when the fractional part is close to $0$; then propitious patterns of $n$ up steps and $n$ down steps can decrease the record a bit. While the fractional part is near $0$, no parity changes can occur. For a parity change to occur, the fractional part has to drift almost by $1$, and that takes longer and longer the further you go. Note that there are relatively large jumps in the last column when the parity of the first column flips. This happens when the fractional part has drifted by about $1-2x$, where $x$ is the current record distance from $\pi/4$.
I don't know about the distribution, or how best to characterize it, but some of your other questions are related to the same phenomenon.
P.S.: This description is only accurate during the stretches of primes of the same order of magnitude. After no improvement has occurred e.g. from $62992$ to $3535871$ (because the process has been wandering around away from $\pi/4$), the step size has of course decreased so much that any memory of "fractional" and "integer" parts from the time of the last improvement has long been erased. Thus at the boundaries between such stretches there is a $50/50$ chance for a parity change.