Rational 8152026

Marvin Burns

August 2026

A New Theoretical Direction from the MRB Computations

This notebook organizes the main ideas in the note into a clean research-style structure, with explanatory text, Wolfram Language scaffolding, and places to insert computations.

1. Overview

The central theme is a possible route toward proving irrationality of the MRB constant, motivated by high-precision computation and by the algebraic structure hidden inside prime radicals, Shanks acceleration, Padé approximation, and Hankel determinants.

The core claims to organize are:

◼

each new prime radical

1/p

contributes genuinely new algebraic information;

◼

prime-ending Shanks approximants remain fractional-linear in

1/p

over the earlier radical field;

◼

inverse-Shanks identities lead to a Hankel-determinant formula

1/p



k+1

, p2k+1;

◼

the determinant ratio is expected to satisfy asymptotics of the form

1/p



-p+o(p)

reflecting the logarithmic capacity of $[0,1]$;

◼

under a rationality assumption for the MRB constant, one obtains very small algebraic integers

p

with a distinguished real embedding of size roughly



exp-

log4

+o

;

◼

the remaining obstruction is arithmetic: control of conjugates, heights, and norms.

2. Working assumptions and notation

We treat the MRB constant as a constant numerically defined from a sequence of partial sums and remainders. Replace the placeholders below with the actual formulas used in your project.

ClearAll[mrbTerm,mrbPartialSum,mrbRemainder,mrbSequence,shanksValue,aitkenValue,hankelRatioValue];

Define the MRB term sequence here:

mrbTerm[n_]:=(*insertthenthMRBterm*)

Partial sums and tails:

mrbPartialSum[n_]:=Sum[mrbTerm[j],{j,1,n}]mrbRemainder[n_]:=(*exactornumericalremainderafternterms*)

If you already have a numerical implementation, you can replace the symbolic definitions with memoized versions.

3. Prime radicals as new algebraic information

The first algebraic observation is that for a prime

, the radical

1/p

should not lie in the field generated by the earlier radicals

1/2

1/3

,…,

1/(p-1)

(p-1)

This suggests that adjoining

1/p

creates a genuine degree-

extension.

3.1 Representing radicals in a common number field

The Wolfram Language functions

ToNumberField

AlgebraicNumber

RootReduce

, and

MinimalPolynomial

are useful here.

ClearAll[earlierRadicals,primeRadicalData];earlerRadicals[p_]:=Table[n^(1/n),{n,2,p-1}]primeRadicalData[p_]:=Module[{elts,nf,pr},elts=earlierRadicals[p];pr=p^(1/p);nf=ToNumberField[Append[elts,pr],All];<|"FieldElements"->nf,"PrimeElement"->Last[nf],"MinimalPolynomialOverQ"->MinimalPolynomial[pr,x]|>]

A practical test is to compare the minimal polynomial of

1/p

over

with its characteristic data in a common number field.

Table[{p,MinimalPolynomial[p^(1/p),x]},{p,{2,3,5,7,11}}]

3.2 Conceptual note

Since

-p

is Eisenstein at

, the polynomial is irreducible over

. The notebook can later be extended to test whether

1/p

lies in the field generated by earlier radicals using

ToNumberField

and

RootReduce

4. Shanks acceleration and Padé structure

A major theme is that Shanks acceleration does not destroy the newest prime radical. Instead, a prime-ending Shanks approximant is expected to be a Möbius transform of

1/p

with coefficients in the field generated by earlier radicals.

The classical relation between Shanks transforms, Padé approximants, and Hankel determinants is essential here. See

PadeApproximant

and

HankelMatrix

4.1 Aitken–Shanks transform scaffold

ClearAll[aitkenTransform,shanksTransform];aitkenTransform[{s0_,s1_,s2_}]:=s0-(s1-s0)^2/(s2-2s1+s0)shanksTransform[list_List]/;Length[list]>=3:=NestList[Function[row,Map[aitkenTransform,Partition[row,3,1]]],list,Length[list]-3]

For a sequence of MRB partial sums:

mrbSequence[n_]:=Table[mrbPartialSum[j],{j,1,n}]

(*ExampleuseoncemrbPartialSumisdefined*)(*shanksTransform[mrbSequence[10]]*)

4.2 Padé viewpoint

The Shanks transform of partial sums is closely tied to diagonal Padé approximants. The Wolfram Language has direct support via

PadeApproximant

Example template:

ClearAll[padeFromSeries];padeFromSeries[seriesExpr_,x_,n_]:=PadeApproximant[seriesExpr,{x,0,n}]

You can compare direct Shanks constructions to

PadeApproximant

exactly as in the standard Hankel-determinant examples.

5. Hankel determinants and inverse-Shanks identities

The note suggests that for primes of the form

p2k+1,

one has an exact identity

1/p



k+1

where

belongs to the field generated by the earlier radicals, and $D_k$ is a Hankel determinant built from MRB remainders.

5.1 Generic Hankel determinant builder

ClearAll[hankelDeterminant];hankelDeterminant[data_List,k_Integer?Positive]:=Module[{firstRow,lastColumn},firstRow=Take[data,k];lastColumn=Take[data,{k,2k-1}];Det[HankelMatrix[firstRow,lastColumn]]]

5.2 Determinants from remainders

5.3 Determinant ratios

This is the natural place to test the conjectural identity numerically or symbolically.

6. Capacity of $[0,1]$ and the appearance of 4

6.1 Explanatory remarks

6.2 Numerical experiment template

7. Rationality assumption and tiny algebraic integers

7.1 Symbolic placeholders

7.2 Algebraic number diagnostics

8. Conjugates, heights, and the arithmetic bottleneck

The notebook should distinguish clearly between:

◼

one very small distinguished embedding;

◼

the full set of conjugates;

◼

the field norm, which multiplies all conjugates together;

◼

denominator growth and height growth.

8.1 Conjugate exploration template

A refined version should work in the actual ambient number field and track embeddings more canonically.

9. Partial norms: eliminating radicals one layer at a time

A striking experimental claim is that taking norms with respect to only the newest radicals—roughly the upper half of the radical tower—preserves or even strengthens the smallness before the full norm eventually becomes too large.

9.1 Organizing the radical tower

9.2 Partial field reduction template

9.3 Research questions

◼

Which radicals can be eliminated while preserving the exceptional smallness?

◼

How does the rank of the induced perturbation depend on the eliminated indices?

◼

Can the low-rank structure force slow enough arithmetic growth to contradict the tiny distinguished value?

10. Low-rank perturbation viewpoint

The note indicates that high-index radicals influence only a small corner of the relevant Hankel matrix, hence contribute a low-rank perturbation.

This deserves its own computational section.

10.1 Matrix decomposition scaffold

10.2 Rank experiments

Possible experiments:

11. Suggested computational agenda

8. Construct $Z_p$ under the rationality assumption.9. Compute minimal polynomials, norms, traces, and denominator data.10. Study partial norms and low-rank perturbation structure.

12. Documentation links

Useful Wolfram Language references:

13. Closing summary

This notebook is designed as a research framework rather than a finished proof. Its purpose is to connect:

◼

prime radicals,

◼

Aitken–Shanks acceleration,

◼

Padé approximation,

◼

Hankel determinant identities,

◼

logarithmic capacity,

◼

and algebraic number theory.

The main open objective is now sharply focused:

If you want, I can next turn this into a more polished publication-style notebook with title page, theorem/proposition formatting, and separate sections for Background, Computational Evidence, Exact Algebraic Structure, and Open Problems.

Acknowledgements

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 .

The Wolfram Notebook Assisted designed this notebook.