From: Simon Plouffe <simon.plouffe@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] a strange class of algebraic numbers I have been working on a simple idea that lead to something interesting. Some of you may know that the iteration Z(n+1)= Z(n)^2 + c for some simple c will lead to an algebraic number, in this case, many times to a 4'th degree algebraic number.
--Comments on paper follow. Warning: I'm handicapped by not speaking French. --if it converges (to Z) then Z = Z^2 + c which is a quadratic equation and hence we would get an algebraic number of degree=2. So evidently Simon Plouffe is speaking of something other than mere convergence, perhaps he means converges to a fixed-length attractive cycle, in which case we would generically expect the algebraic numbers that are the cycle elements, each to have degree = 2^k where k is the cycle's length (cardinality). --In the paper, it would help the reader if, whenever you presented one of your cool graphical images, you were to SAY in the caption, what the width & height of the image are, in pixels, i.e, bits. I presume bits in raster order (rightward in rows, row by row going down?). --Also, your first image seems to have 3 colors, green, black, and blue, which means WHAT about the binary bits you are depicting? And the caption says 270 million bits are depicted, which is absurd; I assure you my computer's screen I am using to view this image is not capable of depicting that much information, so I conclude that really, your picture cheats in some manner, i.e. erases a great deal of information. --What I think is interesting about this all, is it demonstrates that the naive guess that some algebraic irrational is going to have "random" bits, is false. Actually for example 1000 / (500 + sqrt(249999)) = = 1.000001000002000005000014000042000132000429001430004862016796058786... in decimal (note Catalan number sequence) so it is well known that "nonrandomness" can happen -- but Plouffe's numbers "stay nonrandom" for much longer than this example does. --I suspect we can explain what is going on using "generating functions." The above Catalan example arose from the Catalan generating function, see my paper http://rangevoting.org/EnvyFree.html which mentions some in section 2. Now if we instead had a generating function F(x) for a different sequence which did not grow exponentially like the Catalan numbers, but rather grew much more slowly, e.g. logarithmically or polynomially, then this kind of example when x was a suitable power of 2 would (in radix 2) produce numbers with nonrandom-looking patterns which extended for a very long way. Observe that ANY algebraic function of x, call it F(x), which has the property that its points of non-analyticity (in the complex x-plane) have modulus=1, will automatically have a Maclaurin series whose coefficients grow at most only polynomially with n. If in fact these coefficients ARE polynomials in n (as opposed to, say, rational functions, which would be unfriendly with binary neatness) then we'd get a "very nonrandom" number. Also, even with exponential coefficient growth not polynomial, if we merely chose x to be very very tiny and binary-friendly, such as x=2^(-65536), again we'd expect very "lacunary" behavior, which would continue for a long way, i.e. at least until the coefficients grew larger than 1/x. And I notice Plouffe is doing just that. Now consider this example: F(x) = 1 + 32 * x^16 * sqrt(x^(-10)+24*sqrt(x^(-4)+4)) Its Maclaurin series is: F(x) = 1+32*x^11+384*x^19+768*x^23-3072*x^27-7680*x^31+23808*x^35+176640*x^39-281088*x^43-3326976* x^47+168192*x^51+70439424*x^55+80647680*x^59-1354681344*x^63-3764190720*x^67+25309811712*x^71+ 118764966912*x^75-404576667648*x^79-3346846172928*x^83+4756985860608*x^87+85558030695936*x^91+ 11334396625920*x^95-2040547018553856*x^99+... the coefficients in this series appear always to be integers (confirmed out to x^500) of both signs, and appear to increase roughly singly-exponentially with n. I presume the fact these coeffs always are integers can be proven by finding a recurrence which they obey. (Lunnon's "number wall" being one way to seek such.) The powers are always 3 mod 4 so it would have been "better" to consider x*F(x) not F(x). But anyhow, this generating function is responsible for Plouffe's example on his page 6. Observe that every point of non-analyticity of F(x) has modulus 1/sqrt(2) = 0.7071067814 or one of these {0.6599289943, 0.6499107662, 0.7141108228} hence I expect the coefficients to grow roughly like (1/0.6499107662)^n and this kind of argument will prove the |coeffs| indeed behave like simply-exponentially growing functions of n. Now consider G(x) = 1 + (1/2) * x^2 * sqrt(2*x^(-2) + 2*sqrt(x^(-4)+16)) which has Maclaurin series G(x) = 1+1*x+2*x^5-10*x^9+84*x^13-858*x^17+9724*x^21-117572*x^25+1485800*x^29 -19389690*x^33+ 259289580*x^37-3534526380*x^41+48932534040*x^45-686119227300*x^49+9723892802904*x^53-\ 139067101832008*x^57+2004484433302736*x^61-29089272078453818*x^65+424672260824486220*x^69-\ 6232570989814602524*x^73+91901608649243484728*x^77-1360850743459951600780*x^81+ 20227837183275796268040*x^85-301706958410170703321400*x^89+4514235708154496146507440*x^93-\ 67737547514382093772858980*x^97+... again with all coefficients apparently integers of both signs, again growing singly-exponentially. The powers are 1 mod 4 so that G(x)*x^3 would have been "better" than G(x). Anyhow, this generating function is responsible for Plouffe's equation 1 on his page 2. --Warren D. Smith.