MRB Irrationality Trifoliated Relation

Core infographic

Reading the infographic

The infographic summarizes a proposed equivalence connecting the irrationality of the MRB constant to a denominator-growth condition for rationals trapped between consecutive partial sums.
It presents three statements:
1.
C
MRB
∉Q
2.
B
k
∞
3.
∀q≥1, ∃k: ⌊q
S
2k+1
⌋⌊q
S
2k+2
⌋
where
is the
th partial sum of the MRB series.

Definitions

MRB partial sums

Define
These alternating partial sums form nested trapping intervals
which are intended to squeeze toward the MRB constant.

The denominator parameter

For each
, let $B_k$ denote the smallest denominator
for which there exists a reduced rational number
So $B_k$ measures how simple the simplest rational inside the
th trapping interval can be.
◼
  • small $B_k$ means a simple rational lies in the interval
    • ◼
  • large $B_k$ means every rational in the interval needs a more complicated denominator
    • Thus the statement
    means the trapping intervals eventually avoid all rationals with bounded denominator.

    Floor formulation

    The infographic also gives the equivalent condition
    This says that for every fixed integer scaling factor
    , some trapping interval is so short that the two scaled endpoints fall in the same unit interval.
    That is exactly what prevents a rational with denominator
    from lying strictly between them.

    Why the equivalence is plausible

    A reduced rational number with denominator
    has the form
    Such a rational lies in an interval $(x,y)$ exactly when there is an integer
    such that
    Multiplying by
    gives
    So if
    , then there is no integer strictly between
    and
    , hence no rational with denominator
    in the interval.
    Applied to
    the floor condition says that denominator
    is excluded from the
    th trapping interval.
    If this happens for every fixed
    once
    is chosen appropriately, then the smallest admissible denominator must tend to infinity.

    Connection to irrationality

    If a real number
    is rational, say
    in lowest terms, then every sufficiently small interval containing
    also contains a rational with denominator
    —namely
    itself.
    So if the trapping intervals
    shrink to a limit and yet eventually contain no rational of any fixed denominator, that behavior is consistent with the limit being irrational.
    The infographic encapsulates this idea as

    Wolfram Language exploration

    Define the partial sums

    In[]:=
    mrbPartialSum[n_Integer?Positive]:=Sum[(-1)^j(j^(1/j)-1),{j,1,n}]

    First few partial sums

    In[]:=
    Table[{n,N[mrbPartialSum[n],30]},{n,1,12}]
    Out[]=
    {{1,0},{2,0.414213562373095048801688724210},{3,-0.0280360079343133335199495865704},{4,0.386177554438781715281739137639},{5,0.00644789297756688289167567342327},{6,0.354454047574844550265911987326},{7,0.0339847998187207584565846558261},{8,0.330824354469730424390338773619},{9,0.0543063474604887647059401230358},{10,0.313231759254655975129894229432},{11,0.0696565313422852400501025175221},{12,0.299732036920256493713108013639}}

    Consecutive trapping intervals

    In[]:=
    Table[​​{k,N[mrbPartialSum[2k+1],20],N[mrbPartialSum[2k+2],20]},​​{k,0,8}​​]
    Out[]=
    {{0,0,0.41421356237309504880},{1,-0.028036007934313333520,0.38617755443878171528},{2,0.0064478929775668828917,0.35445404757484455027},{3,0.033984799818720758457,0.33082435446973042439},{4,0.054306347460488764706,0.31323175925465597513},{5,0.069656531342285240050,0.29973203692025649371},{6,0.081617993359488297054,0.28906002077789960746},{7,0.091199962508263363034,0.28040707751098442975},{8,0.099055002885438708212,0.27324225598510023772}}

    Interval widths

    In[]:=
    Table[​​{k,N[mrbPartialSum[2k+2]-mrbPartialSum[2k+1],30]},​​{k,0,12}​​]
    Out[]=
    {{0,0.414213562373095048801688724210},{1,0.414213562373095048801688724210},{2,0.348006154597277667374236313903},{3,0.296839554651009665933754117792},{4,0.258925411794167210423954106396},{5,0.230075505577971253663005496117},{6,0.207442027418411310407469876169},{7,0.189207115002721066717499970560},{8,0.174187253099661529512332325896},{9,0.161586349641542281808721224246},{10,0.150851300358278785424559796237},{11,0.141586440632163424460829365197},{12,0.133501376458707753420573350481}}
    Since
    the width is exactly the positive term added at the even step.

    Search for the smallest denominator in a trapping interval

    In[]:=
    clearDenominatorSearch[k_Integer?NonNegative,maxDen_Integer?Positive]:=Module[​​{left,right},​​left=N[mrbPartialSum[2k+1],50];​​right=N[mrbPartialSum[2k+2],50];​​SelectFirst[​​Range[maxDen],​​Function[q,​​AnyTrue[​​Range[Ceiling[qleft],Floor[qright]],​​CoprimeQ[#,q]&&left<#/q<right&​​]​​],​​Missing["NotFound"]​​]​​]

    Floor test for a fixed denominator

    Visualize the trapping intervals and candidate rationals

    Remarks

    ◼
  • The image is an excellent high-level summary of the denominator-exclusion viewpoint.

    • Related documentation

    Useful symbols: