factor PI into two doubles A*B

foo.txt

(*
Pi correctly rounded to 106 digits via sollya:
   > display=dyadic!;
   > round(Pi,106,RN);
   63719069007931157819013617823235b-104
*)

s = 63719069007931157819013617823235

(* prime factors of 's' and nearby integers *)
f0 = FactorInteger[s]
f1 = FactorInteger[s + 1]
f2 = FactorInteger[s + 2]
f3 = FactorInteger[s + 3]
f4 = FactorInteger[s - 1]
f5 = FactorInteger[s - 2]
f6 = FactorInteger[s - 3]

(* output:
    {{5, 1}, {7, 1}, {23, 1}, {29, 1}, {41, 1}, {41385037, 1}, {1608601017035651239, 1}}
    {{2, 2}, {81761, 1}, {194833322146045051488526369, 1}}
    {{3, 3}, {811, 1}, {2909945152666171522081272221, 1}}
    {{2, 1}, {1344138743, 1}, {23702563942809867291733, 1}}
    {{2, 1}, {3, 1}, {13, 1}, {816911141127322536141200228503, 1}}
    {{17, 1}, {74509, 1}, {2874196414547, 1}, {17502308010263, 1}}
    {{2, 9}, {11, 1}, {61, 1}, {263, 1}, {44131, 1}, {29700157, 1}, {538045097321, 1}}
*)

(* eye-ball bit requirements of largest prime *)
{N[Log2[f0[[Length[f0], 1]]]],
 N[Log2[f1[[Length[f1], 1]]]],
 N[Log2[f2[[Length[f2], 1]]]],
 N[Log2[f3[[Length[f3], 1]]]],
 N[Log2[f4[[Length[f4], 1]]]],
 N[Log2[f5[[Length[f5], 1]]]],
 N[Log2[f6[[Length[f6], 1]]]]}

(* output: {60.4805, 87.3324, 91.2331, 74.3275, 99.3661, 43.9926, 38.9689} 
   so only 'f5' & 'f6' have a largest prime that will fit in 53 bits.
   
   Gro-Tsen's constants are 'f6' so (s-3).
   
   (s-2) = 17 * 74509 * 2874196414547 * 17502308010263
   
   no pairs of (s-2) allow both to fit in 53 bits so (s-3) is optimal.
*) 

{ N[Log2[17502308010263 74509]], N[Log2[17 2874196414547]] }
{ N[Log2[17502308010263 17]],    N[Log2[74509 2874196414547]] }

(*
    {60.1777, 45.4738}
    {48.0801, 57.5714}
*)