The Greedy MRB
The Greedy MRB
MarvinRayBurns
mrb=the MRB constant
mrb=the MRB constant
mrb=NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->110];
Introduction
Introduction
Introduction
This notebook investigates a remarkably rapidly convergent approximation to the MRB constant
C_MRB = Sum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}].
The starting point is the residue-magnitude family
A(m) = (2/π) m^(-1/m) Sin[π/m], m ≥ 2,
which arises naturally from the reciprocal-kernel residue structure associated with the MRB constant. The first particularly striking approximation occurs at m = 8,
A(8) = 2^(5/8) Sin[π/8]/π,
for which
C_MRB - A(8) ≈ -1.89180712788007352705453393844*10^-8.
This unexpectedly close approximation suggests a natural question: can the remaining error itself be successively approximated by additional members of the same residue family?
The notebook develops such a greedy residue construction. Given a residual E, the next integer m is chosen so that A(m), with the appropriate sign, is as close as possible to E. Thus an approximation S_k is successively corrected according to
S_(k+1) = S_k + Sign[E_k] A(m_(k+1)),
E_k = C_MRB - S_k.
The essential numerical observation is that
A(m) ~ 2/m,
so a residual of magnitude |E| initially suggests an index near
m ~ 2/|E|.
A more accurate inversion of the asymptotic expansion leads to the continuous predictor
y = 2/|E|,
m_pred = y - Log[y] + (-(1/2) (Log[y])^2 + Log[y] - π^2/6)/y,
after which only a small neighborhood of integers around m_pred needs to be tested.
The resulting indices grow extraordinarily quickly. In the computation below they begin
8, 105719004, 77401040908104410, 24845630402060641702669818431121505, ...
while the corresponding residuals decrease from approximately 10^-8 to 10^-17, 10^-35, 10^-69, 10^-140, and finally about 10^-281 after only six terms.
This behavior is not merely numerical coincidence. Since the spacing between successive residue magnitudes satisfies asymptotically
A(m) - A(m + 1) ~ 2/m^2,
choosing the nearer of two adjacent residue magnitudes produces an approximately quadratic reduction of the residual,
|E_(k+1)| ≤ (1/4 + o(1)) |E_k|^2.
The purpose of this notebook is therefore threefold: to discover the greedy residue pattern computationally, to develop an asymptotic formula that predicts the enormous indices efficiently, and to illustrate the resulting near-quadratic convergence numerically. The final computation uses a 400-digit value of C_MRB and the refined predictor to construct and display the first six greedy terms.
The construction should not, by itself, be interpreted as a proof of irrationality of C_MRB. Rather, it exposes a new and highly structured approximation of the constant by signed quantities of the form
α_m/π, α_m = 2 m^(-1/m) Sin[π/m],
with α_m algebraic. The unusually rapid decay of the greedy residuals makes this representation potentially relevant to the separate question of whether transcendence or irrationality estimates can ultimately be brought to bear on C_MRB.
Introduction
This notebook investigates a remarkably rapidly convergent approximation to the MRB constant
C_MRB = Sum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}].
The starting point is the residue-magnitude family
A(m) = (2/π) m^(-1/m) Sin[π/m], m ≥ 2,
which arises naturally from the reciprocal-kernel residue structure associated with the MRB constant. The first particularly striking approximation occurs at m = 8,
A(8) = 2^(5/8) Sin[π/8]/π,
for which
C_MRB - A(8) ≈ -1.89180712788007352705453393844*10^-8.
This unexpectedly close approximation suggests a natural question: can the remaining error itself be successively approximated by additional members of the same residue family?
The notebook develops such a greedy residue construction. Given a residual E, the next integer m is chosen so that A(m), with the appropriate sign, is as close as possible to E. Thus an approximation S_k is successively corrected according to
S_(k+1) = S_k + Sign[E_k] A(m_(k+1)),
E_k = C_MRB - S_k.
The essential numerical observation is that
A(m) ~ 2/m,
so a residual of magnitude |E| initially suggests an index near
m ~ 2/|E|.
A more accurate inversion of the asymptotic expansion leads to the continuous predictor
y = 2/|E|,
m_pred = y - Log[y] + (-(1/2) (Log[y])^2 + Log[y] - π^2/6)/y,
after which only a small neighborhood of integers around m_pred needs to be tested.
The resulting indices grow extraordinarily quickly. In the computation below they begin
8, 105719004, 77401040908104410, 24845630402060641702669818431121505, ...
while the corresponding residuals decrease from approximately 10^-8 to 10^-17, 10^-35, 10^-69, 10^-140, and finally about 10^-281 after only six terms.
This behavior is not merely numerical coincidence. Since the spacing between successive residue magnitudes satisfies asymptotically
A(m) - A(m + 1) ~ 2/m^2,
choosing the nearer of two adjacent residue magnitudes produces an approximately quadratic reduction of the residual,
|E_(k+1)| ≤ (1/4 + o(1)) |E_k|^2.
The purpose of this notebook is therefore threefold: to discover the greedy residue pattern computationally, to develop an asymptotic formula that predicts the enormous indices efficiently, and to illustrate the resulting near-quadratic convergence numerically. The final computation uses a 400-digit value of C_MRB and the refined predictor to construct and display the first six greedy terms.
The construction should not, by itself, be interpreted as a proof of irrationality of C_MRB. Rather, it exposes a new and highly structured approximation of the constant by signed quantities of the form
α_m/π, α_m = 2 m^(-1/m) Sin[π/m],
with α_m algebraic. The unusually rapid decay of the greedy residuals makes this representation potentially relevant to the separate question of whether transcendence or irrationality estimates can ultimately be brought to bear on C_MRB.
Finding greedy algorithms
Finding greedy algorithms
In[]:=
mrb-sin
5/8
2
π
8
π
Out[]=
-1.8918071278800735270545339384419831105875046368977276124195891210331759512869030718208453771905709658×
-8
10
A very nice way to characterize the m = 8 phenomenon : the residue - magnitude curve is locally almost antisymmetric about the MRB level at x = 8, with the symmetry defect equal, to leading order, to the tiny 1.89*10^−8 discrepancy .
In[]:=
Clear[x,A,err];A[x_]:=(2/Pi)x^(-1/x)Sin[Pi/x];err[x_]:=mrb-A[x];Table[{h,N[err[8-h],30],N[err[8+h],30],N[(err[8-h]+err[8+h])/2,30]},{h,{1/10,1/100,1/1000,1/10000,1/100000,1/1000000}}]//Prepend[#,{"h","E(8-h)","E(8+h)","Symmetric Mean"}]&//TableForm
Out[]//TableForm=
h | E(8-h) | E(8+h) | Symmetric Mean |
1 10 | -0.00192960917827337536029414717019 | 0.00188966419924641899703494329327 | -0.0000199724895134781816296019384574 |
1 100 | -0.000191161375110304434868190258942 | 0.000190724510403526009971859125794 | -2.18432353389212448165566573645× -7 10 |
1 1000 | -0.0000191151867636664558590825909543 | 0.0000190733603397527116678794364611 | -2.09132119568720956015772466187× -8 10 |
1 10000 | -1.92836535713236495344515757346× -6 10 | 1.89048931176163066185201862401× -6 10 | -1.89380226853671457965694747266× -8 10 |
1 100000 | -2.09861004216842198355354979074× -7 10 | 1.72024462631109442464358878048× -7 10 | -1.89182707928663779454980505130× -8 10 |
1 1000000 | -3.80123466163182497757846636939× -8 10 | 1.76200068435466385480992208179× -10 10 | -1.89180732739413916951518357429× -8 10 |
In[]:=
mrb-sin-2sin/.x->105719003
5/8
2
π
8
-1/x
x
π
x
π
Out[]=
1.53107267569582710627152959390974769077146441668694892893106197661120385817703017263828780513×
-16
10
In[]:=
a=8;b=105719003;c=13062737202145246;mrb--+
2Sin
-1/a
a
π
a
π
2Sin
-1/b
b
π
b
π
2Sin
-1/c
c
π
c
π
Out[]=
7.463818455090338325826447591991425342586755915668527118376200714477659264274×
-33
10
In[]:=
Clear[a,b,c]
In[]:=
((2/Pi)a^(-1/a)Sin[Pi/a]-(2/Pi)b^(-1/b)Sin[Pi/b]+(2/Pi)c^(-1/c)Sin[Pi/c])
Out[]=
2Sin
-1/a
a
π
a
π
2Sin
-1/b
b
π
b
π
2Sin
-1/c
c
π
c
π
Finding a pattern
Finding a pattern
In[]:=
mrb=N[NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->210],200];
In[]:=
ClearAll[k,y,pred1,pred2];Grid[Prepend[Table[y=N[2/Abs[errs[[k]]],80];pred1=N[y-Log[y],80];pred2=N[y-Log[y]+Log[y]/y,80];{k,N[pred1,35],N[pred2,35],indices[[k]],N[indices[[k]]-pred1,25],N[indices[[k]]-pred2,25]},{k,1,Length[errs]}],{"Step","y - Log[y]","y - Log[y] + Log[y]/y","Chosen m","Chosen - Pred1","Chosen - Pred2"}],Frame->All,Alignment->Left]
Out[]=
Step | y - Log[y] | y - Log[y] + Log[y]/y | Chosen m | Chosen - Pred1 | Chosen - Pred2 |
1 | 1.0571900385560402266366275328996105× 8 10 | 1.0571900385560419743159942546556093× 8 10 | 105719003 | -0.8556040226636627532899611 | -0.8556041974315994254655609 |
2 | 1.3062737202145245363205283778906634× 16 10 | 1.3062737202145245363205283778909475× 16 10 | 13062737202145246 | 0.63679471622109336575118 | 0.63679471622109052495759 |
3 | 2.6795935780511866700090517370263919× 32 10 | 2.6795935780511866700090517370263919× 32 10 | 267959357805118667000905173702639 | -0.1907207 | -0.1907207 |
Using the pattern
Using the pattern
In[]:=
mrb=N[NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->410],400];
In[]:=
Precision[mrb]
Out[]=
400.
In[]:=
ClearAll[A,predictM,nextIndex,greedyMRB,mrbN,wp,searchRadius,startApprox,rows,displayRows,fmt];wp=350;searchRadius=10;mrbN=N[mrb,wp];(*Positiveresiduemagnitude*)A[m_Integer?Positive]:=N[(2/Pi)m^(-1/m)Sin[Pi/m],wp];(*Refinedcontinuouspredictor*)predictM[err_?NumericQ]:=Module[{y0},y0=N[2/Abs[err],wp];N[y0-Log[y0]+(-(Log[y0]^2)/2+Log[y0]-Pi^2/6)/y0,wp]];(*Initialcontour-derivedm=8approximation*)startApprox=N[(2^(5/8)Sin[Pi/8])/Pi,wp];(*Findthenextgreedyintegernearthepredictor*)nextIndex[err_?NumericQ]:=Module[{sgn,mpred,cand,scored},sgn=Sign[err];mpred=predictM[err];cand=Range[Max[2,Floor[mpred]-searchRadius],Ceiling[mpred]+searchRadius];scored=Table[{m,Abs[err-sgnA[m]]},{m,cand}];First@First@SortBy[scored,Last]];(*Greedyconstruction*)greedyMRB[nTerms_Integer?Positive]:=Module[{approx,err,m,sgn,mpred,newErr,data={}},approx=startApprox;err=N[mrbN-approx,wp];AppendTo[data,{1,8,1,Missing["NotApplicable"],err,Abs[err]}];Do[sgn=Sign[err];mpred=predictM[err];m=nextIndex[err];approx=N[approx+sgnA[m],wp];newErr=N[mrbN-approx,wp];AppendTo[data,{k,m,sgn,mpred,newErr,Abs[newErr]}];err=newErr;,{k,2,nTerms}];data];(*Computesixterms*)rows=greedyMRB[6];(*Displayonlyabout30significantdigitsinapproximatecolumns*)ClearAll[fmt,formatRow];fmt[x_?NumericQ]:=ScientificForm[N[x,30],30];formatRow[row_List]:={row[[1]],(*Term*)row[[2]],(*exactintegerm*)row[[3]],(*Sign*)If[MissingQ[row[[4]]],"",fmt[row[[4]]]],fmt[row[[5]]],(*Residual*)fmt[row[[6]]](*AbsoluteError*)};displayRows=formatRow/@rows;Grid[Prepend[displayRows,{"Term","m","Sign","Predicted m*","Residual","Absolute Error"}],Frame->All,Alignment->Left]
AI contribution
AI contribution
This investigation developed through a combination of analytic calculation, high - precision numerical experimentation, and symbolic verification . ChatGPT, developed by OpenAI, was used interactively during the development of this work
to assist in exploring contour representations, branch - cut transformations, residue calculations, asymptotic expansions, and related analytic questions, as well as in the preparation and revision
of portions of the manuscript . Wolfram Mathematica was used for high - precision numerical computation and symbolic verification where appropriate . All mathematical claims, derivations, numerical results, and conclusions presented in the final
manuscript were reviewed and accepted by the author, who assumes full responsibility for the
content of the work .
to assist in exploring contour representations, branch - cut transformations, residue calculations, asymptotic expansions, and related analytic questions, as well as in the preparation and revision
of portions of the manuscript . Wolfram Mathematica was used for high - precision numerical computation and symbolic verification where appropriate . All mathematical claims, derivations, numerical results, and conclusions presented in the final
manuscript were reviewed and accepted by the author, who assumes full responsibility for the
content of the work .
This is from a collaboration from the Author and GPT-5.6 Sol. The inspiration was an intuition the author had that such an algorithm existed and would follow such a simple pattern.