April Fool’s day 2021 was near the start of π’s continued fraction
​By Bill Gosper
There are 38 2021s in the first hundred million terms of π
In[]:=
positions=Position[pi=ContinuedFraction[π,99999999],2021]
NumberLinePlot[Flatten[positions]]
Out[]=
with six among the first ten million:
Take[pi,#[[1]]-{2,0}]&/@Take[positions,6]
Out[]=
{{1,1,2021},{14,3,2021},{3,1,2021},{20,8,2021},{1,1,2021},{3,1,2021}}
The elements in the form (x, y, 2021) are
In[]:=
dates=DeleteDuplicates[Take[pi,#[[1]]-{2,0}]&/@positions]
Out[]=
{{1,1,2021},{14,3,2021},{3,1,2021},{20,8,2021},{8,1,2021},{1,55,2021},{1,28,2021},{27,14,2021},{2,1,2021},{7,2,2021},{4,4,2021},{1,2,2021},{2,2,2021},{1,3,2021},{1,7,2021},{1,4,2021},{19,1,2021},{1,8,2021},{1,30,2021},{240,1,2021},{6,1,2021},{4,1,2021},{25,1,2021},{5,1,2021},{27,1,2021}}
Notice that nearly all of these serve for the interpretation (day, month, year) and few work for (month, day, year). I.e., (large, small, 2021) is much more probable than (small, large, 2021). This is predicted by the Gauss-Kuzmin distribution, which can predict the probability of any sequence of terms, not just individual terms:
In[]:=
cfprob[L_List]:=Abs@Log[2,(1+1/FromContinuedFraction@L)/(1+1/FromContinuedFraction@MapAt[#+1&,L,-1])]
In[]:=
cfprob[r:(Integer_|Rational_)]:=cfprob@ContinuedFraction@r
Thus, (31, Jan, 2021) is nearly twice as likely as (Jan, 31, 2021):
In[]:=
cfprob/@{{1,31,2021},{31,1,2021}}//N
Out[]=
{1.75115×
-10
10
,3.34×
-10
10
}
For April Fool’s Day,
In[]:=
cfprob/@{{1,4,2021},{4,1,2021}}//N
Out[]=
{7.84375×
-9
10
,1.17586×
-8
10
}
All of this hinges on the unproven but widely believed conjecture that π’s continued fraction is typical of real numbers, and obeys Gauss-Kuzmin.