Factorial Reduction

Author

Eric W. Weisstein
October 3, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/FactorialReduction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FactorialReduction.html.

©2023 Wolfram Research, Inc. except for portions noted otherwise

Source

https://arxiv.org/pdf/2308.11829.pdf

Examples

UnreducedConvergents[a_,b_,{n_,nmax_}]:=Module[
{p,q},
{
RecurrenceTable[{p[n]==a p[n-1]+b p[n-2],p[-1]==1,p[0]==(a/.n->0)},p,{n,1,nmax}],
RecurrenceTable[{q[n]==a q[n-1]+b q[n-2],q[-1]==0,q[0]==1},q,{n,1,nmax}]
}
]

(6/(Zeta[3]))==5+(Underoverscript[\[ContinuedFractionK], (n=1), ∞])((-(n^6))/(17 ((n^3)+(((n+1))^3))-12 (2 n+1)))

5+ContinuedFractionK[-(n^6),17((n^3)+(((n+1))^3))-12(2n+1),{n,∞}]

cf=5+ContinuedFractionK[-(n^6),17((n^3)+(((n+1))^3))-12(2n+1),{n,100}];

Table[17((n^3)+(((n+1))^3))-12(2n+1),{n,0,10}]

N[cf-(6/(Zeta[3])),1000]//N

convergents=Transpose[Abs[NumeratorDenominator[Convergents[(6/(Zeta[3]))-5,20]]]];

With[{
opts=Sequence@@{ImageSize->Medium,LabelStyle->Directive[Black,FontFamily->"Times",14]},
orange=,
blue=
},
GraphicsRow[{
ListLogPlot[
UnreducedConvergents[-(n^6),17((n^3)+(((n+1))^3))-12(2n+1),{n,20}],
AxesLabel->{n,{Style[Abs[(Subscript[p, n])],orange],Style[(Subscript[q, n]),blue]}},
opts
],
ListLogPlot[
convergents,
AxesLabel->{n,{Style[Abs[(Subscript[p, n])]/GCD[(Subscript[p, n]),(Subscript[q, n])],orange],Style[(Subscript[q, n])/GCD[(Subscript[p, n]),(Subscript[q, n])],blue]}},
opts
]
}]]