Re: [math-fun] a strange class of algebraic numbers
Julian, do you see any way to lengthen the nonrandomness? --Bill On Wed, Mar 19, 2014 at 5:12 PM, Julian Ziegler Hunts <julianj.zh@gmail.com>wrote:
What's strange about this? Other than the long runs of identical bits, which are easily explained by rewriting it as 1+Sqrt[1+(Sqrt[1+4^(1-k)]-1)/2]/2^(k/2+1/2) (which accurately predicts a lack of structure for k even), and noting that the series coefficients for sqrt(1+x) have power-of-two denominators (and therefore so do the coefficients of sqrt(1+(sqrt(1+x)-1)/2)).
Julian
On Tue, Mar 18, 2014 at 4:52 PM, Bill Gosper <billgosper@gmail.com> wrote:
Nice. But your plots are line-wrapped. Out[698]= 1 + 2^(-1 - k) Sqrt[2^k + 2 Sqrt[1 + 2^(2 (-1 + k))]]
You have, e.g. In[700]:= ArrayPlot[ Partition[RealDigits[%698 /. k -> 63, 2, 63^2][[1]], 63]]
Try In[705]:= ArrayPlot[ Partition[ Join[ConstantArray[0, 32], RealDigits[%698 /. k -> 63, 2, 2*63^2][[1]]], 2*63]] --rwg Violating two embargoes at once: Climate change makes planes disappear.
Hello, the function you mention is not the same, 1+Sqrt[1+(Sqrt[1+4^(1-k)]-1)/2]/2^(k/2+1/2) the original one posted is : 1 + 1/4*(2*4^s + 2*(16^s + 1)^(1/2))^(1/2)/(2^s)^2 which pushes the pattern very far when (here s) is equal to 100,1000 or 1000000, far : many billion bits. The one you mention only pushes up to a few hundreds bits I could see. What type of gen. function would push the coefficients up to the billlions without being too big ? I don't see. Best regards, Simon Plouffe
The two functions are the same, after replacing k by 2s+1. Even k/half-integral s only produce a little bit of regularity, namely the first few bits because it's very close to 1, but odd k/integral s produce the longer (length quadratic in k) patterns.
do you see any way to lengthen the nonrandomness?
Find some other algebraic function whose Taylor series' coefficients have power-of-two denominators and numerators that grow more slowly (sub-exponentially, if possible)? The actual size of the denominators doesn't matter too much, unless they grow really quickly, since only the ratio of two consecutive denominators contributes (currently, the denominators grow exponentially, hence contribute only a constant (because the numerators are also exponential) amount to the non-randomness). I don't know how you would go about constructing such a function, though. Julian On Thu, Mar 27, 2014 at 2:43 AM, Simon Plouffe <simon.plouffe@gmail.com>wrote:
Hello, the function you mention is not the same,
1+Sqrt[1+(Sqrt[1+4^(1-k)]-1)/2]/2^(k/2+1/2)
the original one posted is : 1 + 1/4*(2*4^s + 2*(16^s + 1)^(1/2))^(1/2)/(2^s)^2
which pushes the pattern very far when (here s) is equal to 100,1000 or 1000000, far : many billion bits. The one you mention only pushes up to a few hundreds bits I could see.
What type of gen. function would push the coefficients up to the billlions without being too big ? I don't see.
Best regards,
Simon Plouffe
________________________________ From: Julian Ziegler Hunts <julianj.zh@gmail.com> To: Simon Plouffe <simon.plouffe@gmail.com> Cc: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, March 27, 2014 11:40 AM Subject: Re: [math-fun] a strange class of algebraic numbers
The two functions are the same, after replacing k by 2s+1. Even k/half-integral s only produce a little bit of regularity, namely the first few bits because it's very close to 1, but odd k/integral s produce the longer (length quadratic in k) patterns.
do you see any way to lengthen the nonrandomness?
Find some other algebraic function whose Taylor series' coefficients have power-of-two denominators and numerators that grow more slowly (sub-exponentially, if possible)? The actual size of the denominators doesn't matter too much, unless they grow really quickly, since only the ratio of two consecutive denominators contributes (currently, the denominators grow exponentially, hence contribute only a constant (because the numerators are also exponential) amount to the non-randomness). I don't know how you would go about constructing such a function, though.
Julian --------------------------------------------- The series sum(x^k/2^k, k=0..inf) = 1/(1 - (x/2)) satisfies Julian's requirement.
-- Gene
participants (4)
-
Bill Gosper -
Eugene Salamin -
Julian Ziegler Hunts -
Simon Plouffe