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.

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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  contributes genuinely new algebraic information;

prime-ending Shanks approximants remain fractional-linear in  over the earlier radical field;

inverse-Shanks identities lead to a Hankel-determinant formula

the determinant ratio is expected to satisfy asymptotics of the form

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

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

with a distinguished real embedding of size roughly

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

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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_]:=(* insert the nth MRB term *)

Partial sums and tails:

mrbPartialSum[n_]:=Sum[mrbTerm[j],{j,1,n}]
mrbRemainder[n_]:=(* exact or numerical remainder after n terms *)

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

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3. Prime radicals as new algebraic information

The first algebraic observation is that for a prime , the radical  should not lie in the field generated by the earlier radicals This suggests that adjoining  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  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  is Eisenstein at , the polynomial is irreducible over . The notebook can later be extended to test whether  lies in the field generated by earlier radicals using ToNumberField and RootReduce.

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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  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/(s2-2 s1+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}]

(* Example use once mrbPartialSum is defined *)
(* 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.

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5. Hankel determinants and inverse-Shanks identities

The note suggests that for primes of the form one has an exact identity 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,2 k-1}];
  Det[HankelMatrix[firstRow,lastColumn]]
]

5.2 Determinants from remainders

ClearAll[mrbRemainderList,mrbHankelD];

mrbRemainderList[n_]:=Table[mrbRemainder[j],{j,1,n}]

mrbHankelD[k_]:=hankelDeterminant[mrbRemainderList[2 k-1],k]

5.3 Determinant ratios

ClearAll[hankelRatio];

hankelRatio[k_]:=mrbHankelD[k+1]/mrbHankelD[k]

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

Table[
  {k,hankelRatio[k]},
  {k,1,6}
]

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

The explanation proposed in the note is potential-theoretic: the remainders admit a signed moment representation on $[0,1]$, and the relevant Padé–Hankel asymptotics are therefore governed by the logarithmic capacity This leads naturally to an exponential scale

6.1 Explanatory remarks

If the remainder sequence behaves like moments of a measure supported on $[0,1]$, then the asymptotics of Padé denominators and Hankel determinants are often controlled by the compact set supporting that measure. For the interval $[0,1]$, capacity rescales from the standard fact to

6.2 Numerical experiment template

Once  is available numerically, test whether

ClearAll[xiP,capacityExperiment];

xiP[p_]:=(* insert the field-valued approximant using earlier radicals *)

capacityExperiment[p_]:=-Log[Abs[p^(1/p)-xiP[p]]]/p

Table[
  {p,N[capacityExperiment[p],30]},
  {p,{3,5,7,11,13,17,19}}
]

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7. Rationality assumption and tiny algebraic integers

Assume, for contradiction, that the MRB constant is rational. Then the note suggests one can derive algebraic integers of the form whose distinguished real value is exceptionally small: This section should keep track of the algebraic data precisely.

7.1 Symbolic placeholders

ClearAll[aK,bK,zP];

aK[k_]:=(* numerator-type quantity A_k *)
bK[k_]:=(* denominator-type quantity B_k *)

zP[p_]:=Module[{k=Quotient[p-1,2]},p bK[k]^p-aK[k]^p]

7.2 Algebraic number diagnostics

Use MinimalPolynomial, AlgebraicNumberNorm, AlgebraicNumberTrace, AlgebraicIntegerQ, and AlgebraicNumberDenominator to study these objects.

ClearAll[zPDiagnostics];

zPDiagnostics[p_]:=Module[{z=zP[p]},
  <|
    "AlgebraicIntegerQ"->AlgebraicIntegerQ[z],
    "Denominator"->AlgebraicNumberDenominator[z],
    "MinimalPolynomial"->MinimalPolynomial[z,x],
    "Norm"->AlgebraicNumberNorm[z],
    "Trace"->AlgebraicNumberTrace[z]
  |>
]

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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

ClearAll[allConjugates];

allConjugates[alg_]:=x /. NSolve[MinimalPolynomial[alg,x]==0,x]

ClearAll[conjugateProfile];

conjugateProfile[alg_]:=Module[{conj=allConjugates[alg]},
  <|
    "Conjugates"->conj,
    "AbsoluteValues"->Abs[conj],
    "MaxAbs"->Max[Abs[conj]],
    "MinAbs"->Min[Abs[conj]]
  |>
]

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

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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.

This suggests studying intermediate norm maps rather than the full norm to .

9.1 Organizing the radical tower

ClearAll[radicalTower];

radicalTower[n_]:=Table[j^(1/j),{j,2,n}]

9.2 Partial field reduction template

ClearAll[toIntermediateField];

toIntermediateField[expr_,generators_List]:=ToNumberField[Append[generators,expr],All]

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?

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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

ClearAll[hankelMatrixFromRemainders,cornerPerturbation];

hankelMatrixFromRemainders[k_]:=HankelMatrix[
  Take[mrbRemainderList[2 k-1],k],
  Take[mrbRemainderList[2 k-1],{k,2 k-1}]
]

cornerPerturbation[base_,perturbed_]:=perturbed-base

10.2 Rank experiments

ClearAll[perturbationRank];

perturbationRank[base_,perturbed_]:=MatrixRank[cornerPerturbation[base,perturbed]]

Possible experiments:

(* Compare Hankel matrices before and after adjoining selected radicals *)

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11. Suggested computational agenda

1. Define the exact MRB term and remainder formulas.2. Build partial sums and Shanks tables.3. Identify the prime-ending approximants explicitly.4. **Express them as fractional-linear functions of .**

5. Construct the inverse-Shanks Hankel determinants $D_k$.6. Test the identity

7. **Measure the slope toward ** in the exponential law.

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.

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12. Documentation links

Useful Wolfram Language references:

PadeApproximant

HankelMatrix

ToNumberField

AlgebraicNumber

MinimalPolynomial

AlgebraicNumberNorm

AlgebraicNumberTrace

AlgebraicIntegerQ

AlgebraicNumberDenominator

RootReduce

RootApproximant

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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:

({{"Determine whether the arithmetic complexity of the low-rank determinant constructions grows slowly enough that the cubic-exponentially small distinguished value of the associated algebraic integers becomes impossible."}})

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.