[math-fun] denesting mumbles
√(1 - ⁿ√R) denests when a certain polynomial P_n(R,x) factors. E.g. P₄(R,x) =1 - 256 x^2 + 128 R x^2 + 1024 R x^4 + 3200 R^2 x^4 - 8192 R^3 x^6 - 28672 R^4 x^8 + 34816 R^5 x^8 - 131072 R^6 x^10 + 131072 R^7 x^12 + 1048576 R^10 x^16 . These grow *fast* with n. I'm really interested in n=6 and 7, for each of which I have one "splitting" R. So I decided to collect a bunch of "toy" R for n=4 from old Macsyma notebooks. I can't find them!! I did find a small one converted to an html. It gave R= 64512/260144641 In[748]:= Factor[%728[[1]] /. R -> %] (using a[3] for x) Out[748]= ((-33038369407 + 458752 a[3]^2) (-33038369407 + 589824 a[3]^2) (1091533853073393531649 + 33569097773154048 a[3]^2 + 262193283072 a[ 3]^4) (1191446152405248657777607437681912764659201 - 304972394845902975940055612302553191173586944 a[3]^2 + 302486329754988497319546160627366515179520 a[3]^4 + 18453680701462988948598008315904 a[ 3]^8))/14195439340812710119888169814159419160980592549659303797\ 19569954760828848317673958401 (How the he<< did I find that?) Ok, let's just try a million R and look for a pattern: Do[If[Mod[i, 10000] == 0, Print[i]]; If[{} =!= #, Print[pos2newman[i], #]] &@ Cases[Factor[%728[[1]] /. R -> pos2newman[i]], p_Plus /; Length[p] == 2], {i, 1, 999999}] // tim 10000 20000 ... 890000 900000 Nada! OK, how long do we need to wait for Mr. Newman to reach 64512/260144641? In[742]:= newman2pos[64512/260144641]//Short Out[742]//Short= 3490308671510526617822104163<<6014>>9495630609150025673335635968 In[743]:= N[%] Out[743]= 3.490308671510527*10^6069 A mere googol^60. Maybe I'm just impatient, but I'd really like a newmanoid bijection with more highly composite rationals. E.g. 69 ↔︎ {1,0,-2,1}, where the list is exponents of 2,3,5,7,... . Or better yet, somebody just tell me how to find the magic Rs. --Bill
Another example for R is 192/2401 which leads to this beauty: sqrt(7 - 2*12^(1/4)) = 1 - sqrt(3) + sqrt(3 + 2*sqrt(3)). There are plenty of such R's. For example, just take R = 16*k^2*(k^2 - 1)/(2*k^2 - 1)^4. My R is from k = 2, and Bill Gosper's is from k = 8. However, I don't know if this covers all R's that factor P_4(R,x). On Sun, Jan 4, 2015 at 12:38 PM, Bill Gosper <billgosper@gmail.com> wrote:
√(1 - ⁿ√R) denests when a certain polynomial P_n(R,x) factors. E.g. P₄(R,x) =1 - 256 x^2 + 128 R x^2 + 1024 R x^4 + 3200 R^2 x^4 - 8192 R^3 x^6 - 28672 R^4 x^8 + 34816 R^5 x^8 - 131072 R^6 x^10 + 131072 R^7 x^12 + 1048576 R^10 x^16 . These grow *fast* with n. I'm really interested in n=6 and 7, for each of which I have one "splitting" R. So I decided to collect a bunch of "toy" R for n=4 from old Macsyma notebooks. I can't find them!! I did find a small one converted to an html. It gave
R= 64512/260144641
In[748]:= Factor[%728[[1]] /. R -> %] (using a[3] for x) Out[748]= ((-33038369407 + 458752 a[3]^2) (-33038369407 + 589824 a[3]^2) (1091533853073393531649 + 33569097773154048 a[3]^2 + 262193283072 a[ 3]^4) (1191446152405248657777607437681912764659201 - 304972394845902975940055612302553191173586944 a[3]^2 + 302486329754988497319546160627366515179520 a[3]^4 + 18453680701462988948598008315904 a[ 3]^8))/14195439340812710119888169814159419160980592549659303797\ 19569954760828848317673958401
(How the he<< did I find that?) Ok, let's just try a million R and look for a pattern: Do[If[Mod[i, 10000] == 0, Print[i]]; If[{} =!= #, Print[pos2newman[i], #]] &@ Cases[Factor[%728[[1]] /. R -> pos2newman[i]], p_Plus /; Length[p] == 2], {i, 1, 999999}] // tim
10000
20000 ... 890000
900000
Nada! OK, how long do we need to wait for Mr. Newman to reach 64512/260144641?
In[742]:= newman2pos[64512/260144641]//Short Out[742]//Short= 3490308671510526617822104163<<6014>>9495630609150025673335635968 In[743]:= N[%] Out[743]= 3.490308671510527*10^6069
A mere googol^60. Maybe I'm just impatient, but I'd really like a newmanoid bijection with more highly composite rationals. E.g. 69 ↔︎ {1,0,-2,1}, where the list is exponents of 2,3,5,7,... . Or better yet, somebody just tell me how to find the magic Rs. --Bill _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Should be further simplified to: sqrt(7 - 2*12^(1/4)) = 1 - sqrt(3) + (3/4)^(1/4) + (27/4)^(1/4). On Mon, Jan 5, 2015 at 7:21 AM, Warut Roonguthai <warut822@gmail.com> wrote:
Another example for R is 192/2401 which leads to this beauty:
sqrt(7 - 2*12^(1/4)) = 1 - sqrt(3) + sqrt(3 + 2*sqrt(3)).
There are plenty of such R's. For example, just take
R = 16*k^2*(k^2 - 1)/(2*k^2 - 1)^4.
My R is from k = 2, and Bill Gosper's is from k = 8.
However, I don't know if this covers all R's that factor P_4(R,x).
On Sun, Jan 4, 2015 at 12:38 PM, Bill Gosper <billgosper@gmail.com> wrote:
√(1 - ⁿ√R) denests when a certain polynomial P_n(R,x) factors. E.g. P₄(R,x) =1 - 256 x^2 + 128 R x^2 + 1024 R x^4 + 3200 R^2 x^4 - 8192 R^3 x^6 - 28672 R^4 x^8 + 34816 R^5 x^8 - 131072 R^6 x^10 + 131072 R^7 x^12 + 1048576 R^10 x^16 . These grow *fast* with n. I'm really interested in n=6 and 7, for each of which I have one "splitting" R. So I decided to collect a bunch of "toy" R for n=4 from old Macsyma notebooks. I can't find them!! I did find a small one converted to an html. It gave
R= 64512/260144641
In[748]:= Factor[%728[[1]] /. R -> %] (using a[3] for x) Out[748]= ((-33038369407 + 458752 a[3]^2) (-33038369407 + 589824 a[3]^2) (1091533853073393531649 + 33569097773154048 a[3]^2 + 262193283072 a[ 3]^4) (1191446152405248657777607437681912764659201 - 304972394845902975940055612302553191173586944 a[3]^2 + 302486329754988497319546160627366515179520 a[3]^4 + 18453680701462988948598008315904 a[ 3]^8))/14195439340812710119888169814159419160980592549659303797\ 19569954760828848317673958401
(How the he<< did I find that?) Ok, let's just try a million R and look for a pattern: Do[If[Mod[i, 10000] == 0, Print[i]]; If[{} =!= #, Print[pos2newman[i], #]] &@ Cases[Factor[%728[[1]] /. R -> pos2newman[i]], p_Plus /; Length[p] == 2], {i, 1, 999999}] // tim
10000
20000 ... 890000
900000
Nada! OK, how long do we need to wait for Mr. Newman to reach 64512/260144641?
In[742]:= newman2pos[64512/260144641]//Short Out[742]//Short= 3490308671510526617822104163<<6014>>9495630609150025673335635968 In[743]:= N[%] Out[743]= 3.490308671510527*10^6069
A mere googol^60. Maybe I'm just impatient, but I'd really like a newmanoid bijection with more highly composite rationals. E.g. 69 ↔︎ {1,0,-2,1}, where the list is exponents of 2,3,5,7,... . Or better yet, somebody just tell me how to find the magic Rs. --Bill _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
For the two examples I got from some math-fun posts sometime ago: sqrt(161 - 12*5^(1/4)) = 2*5^(3/4) - 3*sqrt(5) + 4*5^(1/4) + 6 and sqrt(31 - 4*15^(1/4)) = (3^(1/4)*5^(3/4) - 2*sqrt(5) + 3^(3/4)*5^(1/4) + 2*sqrt(3))/sqrt(2), they corresponds to k = 9 and k = 4, respectively. Let's try a rational k. For k = 3/2, we have sqrt(7 - 2*45^(1/4)) = (sqrt(3) - sqrt(15) + 5^(1/4) + 5^(3/4))/2. In short, sqrt((2k^2-1) - 2*(k^4-k^2)^(1/4)) = (sqrt(k^2-k) - sqrt(k^2+k) + (k+1)^(1/4)*(k-1)^(3/4) + (k-1)^(1/4)*(k+1)^(3/4))/sqrt(2). Beautiful, isn't it? On Mon, Jan 5, 2015 at 9:43 AM, Warut Roonguthai <warut822@gmail.com> wrote:
Should be further simplified to:
sqrt(7 - 2*12^(1/4)) = 1 - sqrt(3) + (3/4)^(1/4) + (27/4)^(1/4).
On Mon, Jan 5, 2015 at 7:21 AM, Warut Roonguthai <warut822@gmail.com> wrote:
Another example for R is 192/2401 which leads to this beauty:
sqrt(7 - 2*12^(1/4)) = 1 - sqrt(3) + sqrt(3 + 2*sqrt(3)).
There are plenty of such R's. For example, just take
R = 16*k^2*(k^2 - 1)/(2*k^2 - 1)^4.
My R is from k = 2, and Bill Gosper's is from k = 8.
However, I don't know if this covers all R's that factor P_4(R,x).
On Sun, Jan 4, 2015 at 12:38 PM, Bill Gosper <billgosper@gmail.com> wrote:
√(1 - ⁿ√R) denests when a certain polynomial P_n(R,x) factors. E.g. P₄(R,x) =1 - 256 x^2 + 128 R x^2 + 1024 R x^4 + 3200 R^2 x^4 - 8192 R^3 x^6 - 28672 R^4 x^8 + 34816 R^5 x^8 - 131072 R^6 x^10 + 131072 R^7 x^12 + 1048576 R^10 x^16 . These grow *fast* with n. I'm really interested in n=6 and 7, for each of which I have one "splitting" R. So I decided to collect a bunch of "toy" R for n=4 from old Macsyma notebooks. I can't find them!! I did find a small one converted to an html. It gave
R= 64512/260144641
In[748]:= Factor[%728[[1]] /. R -> %] (using a[3] for x) Out[748]= ((-33038369407 + 458752 a[3]^2) (-33038369407 + 589824 a[3]^2) (1091533853073393531649 + 33569097773154048 a[3]^2 + 262193283072 a[ 3]^4) (1191446152405248657777607437681912764659201 - 304972394845902975940055612302553191173586944 a[3]^2 + 302486329754988497319546160627366515179520 a[3]^4 + 18453680701462988948598008315904 a[ 3]^8))/14195439340812710119888169814159419160980592549659303797\ 19569954760828848317673958401
(How the he<< did I find that?) Ok, let's just try a million R and look for a pattern: Do[If[Mod[i, 10000] == 0, Print[i]]; If[{} =!= #, Print[pos2newman[i], #]] &@ Cases[Factor[%728[[1]] /. R -> pos2newman[i]], p_Plus /; Length[p] == 2], {i, 1, 999999}] // tim
10000
20000 ... 890000
900000
Nada! OK, how long do we need to wait for Mr. Newman to reach 64512/260144641?
In[742]:= newman2pos[64512/260144641]//Short Out[742]//Short= 3490308671510526617822104163<<6014>>9495630609150025673335635968 In[743]:= N[%] Out[743]= 3.490308671510527*10^6069
A mere googol^60. Maybe I'm just impatient, but I'd really like a newmanoid bijection with more highly composite rationals. E.g. 69 ↔︎ {1,0,-2,1}, where the list is exponents of 2,3,5,7,... . Or better yet, somebody just tell me how to find the magic Rs. --Bill _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Are there extensions of Bernoulli numbers to real (or complex) index? Is there one that is generally accepted as the natural way to do it? It would be nice if there were a natural real-index Bernoulli function that took real values. I read somewhere that Ramanujan used to refer to real order Bernoulli numbers in his notebooks. Does anyone know what definition he used? --Dan
http://arxiv.org/abs/physics/9705021 On Tue, Jan 6, 2015 at 11:16 AM, Daniel Asimov <dasimov@earthlink.net> wrote:
Are there extensions of Bernoulli numbers to real (or complex) index?
Is there one that is generally accepted as the natural way to do it?
It would be nice if there were a natural real-index Bernoulli function that took real values.
I read somewhere that Ramanujan used to refer to real order Bernoulli numbers in his notebooks. Does anyone know what definition he used?
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Thanks, Mike. That's a very nice paper! --Dan
On Jan 6, 2015, at 12:04 PM, Mike Stay <metaweta@gmail.com> wrote:
http://arxiv.org/abs/physics/9705021
On Tue, Jan 6, 2015 at 11:16 AM, Daniel Asimov <dasimov@earthlink.net> wrote:
Are there extensions of Bernoulli numbers to real (or complex) index?
Is there one that is generally accepted as the natural way to do it?
It would be nice if there were a natural real-index Bernoulli function that took real values.
I read somewhere that Ramanujan used to refer to real order Bernoulli numbers in his notebooks. Does anyone know what definition he used?
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
(Redundantly CCing certain funsters because my math-fun is often delayed several hours.) On Sat, Jan 3, 2015 at 9:38 PM, Bill Gosper <billgosper@gmail.com> wrote:
√(1 - ⁿ√R) denests when a certain polynomial P_n(R,x) factors. E.g. P₄(R,x) =1 - 256 x^2 + 128 R x^2 + 1024 R x^4 + 3200 R^2 x^4 - 8192 R^3 x^6 - 28672 R^4 x^8 + 34816 R^5 x^8 - 131072 R^6 x^10 + 131072 R^7 x^12 + 1048576 R^10 x^16 . These grow *fast* with n. I'm really interested in n=6 and 7, for each of which I have one "splitting" R. So I decided to collect a bunch of "toy" R for n=4 from old Macsyma notebooks. I can't find them!! I did find a small one converted to an html. It gave
R= 64512/260144641
In[748]:= Factor[%728[[1]] /. R -> %] (using a[3] for x) Out[748]= ((-33038369407 + 458752 a[3]^2) (-33038369407 + 589824 a[3]^2) (1091533853073393531649 + 33569097773154048 a[3]^2 + 262193283072 a[ 3]^4) (1191446152405248657777607437681912764659201 - 304972394845902975940055612302553191173586944 a[3]^2 + 302486329754988497319546160627366515179520 a[3]^4 + 18453680701462988948598008315904 a[ 3]^8))/14195439340812710119888169814159419160980592549659303797\ 19569954760828848317673958401
(How the he<< did I find that?) Ok, let's just try a million R and look for a pattern: Do[If[Mod[i, 10000] == 0, Print[i]]; If[{} =!= #, Print[pos2newman[i], #]] &@ Cases[Factor[%728[[1]] /. R -> pos2newman[i]], p_Plus /; Length[p] == 2], {i, 1, 999999}] // tim
10000
20000 ... 890000
900000
Nada! OK, how long do we need to wait for Mr. Newman to reach 64512/260144641?
In[742]:= newman2pos[64512/260144641]//Short Out[742]//Short= 3490308671510526617822104163<<6014>>9495630609150025673335635968 In[743]:= N[%] Out[743]= 3.490308671510527*10^6069
A mere googol^60. Maybe I'm just impatient, but I'd really like a newmanoid bijection with more highly composite rationals. E.g. 69 ↔︎ {1,0,-2,1}, where the list is exponents of 2,3,5,7,... .
(Tried another million. More nada.) Found the notebooks! In a Mac folder somehow unshared with XP. Gack! One contains (in effect) Sqrt[1 + (2 Sqrt[b] (-1 + b^2)^(1/4))/(-1 + 2 b^2)] == ((-1 + b)^( 1/4) + (-1 + b)^(1/4) b + (-1 + b)^(3/4) Sqrt[1 + b] + Sqrt[b] (-Sqrt[-1 + b] (1 + b)^(1/4) + (1 + b)^(3/4)))/( b^(1/4) Sqrt[(-2 + 4 b^2)/Sqrt[b/(1 + b)]]) almost identical with Warut Roonguthai <warut822@gmail.com> wrote: Another example for R is 192/2401 which leads to this beauty: sqrt(7 - 2*12^(1/4)) = 1 - sqrt(3) + sqrt(3 + 2*sqrt(3)). There are plenty of such R's. For example, just take R = 16*k^2*(k^2 - 1)/(2*k^2 - 1)^4. My R is from k = 2, and Bill Gosper's is from k = 8. However, I don't know if this covers all R's that factor P_4(R,x) Or better yet, somebody just tell me how to find the magic Rs.
--Bill
Yes, please, Warut, remind me how we got this! --rwg Is there any hope to do this for R^(1/6)? I.e., (six terms)^2 = two terms? The R^(1/4) case can never exceed (four terms)^2 = two terms = 1+R^(1/4) when b (or k) = √rational. --rwg
participants (5)
-
Bill Gosper -
Daniel Asimov -
Daniel Asimov -
Mike Stay -
Warut Roonguthai