[seqfan] Re: A053373 (fundamental unit for real quadratic field...)

[Emmanuel Vantieghem]

Dear Sir,

The inserted values (352, 45, 5640)  are the  y-values of the fundamental
unit of the fields  Q(Sqrt(d))  for  d=153, 261  and  297.
These values of  d  are not squarefree.  So, in my opinion, these values
should not be added to  A053373.

Sincere Greetings.

Emmanuel.
_________________________

2012/12/12 Jean-François Alcover <jf.alcover at gmail.com>

> Dear SeqFans,
>
> Using this Mma program (coded after Max Alexeyev comment) :
> g[n_] := Ceiling[Sqrt[n]] + n - 1;
>
> r[n_] := Reduce[ an > 0 && (bn^2 - bn*an - g[n]*an^2 == 1 ||
> bn^2 - bn*an - g[n]*an^2 == -1), {an, bn}, Integers] /. C -> c;
>
> ab[n_] := DeleteCases[ Flatten[Table[{an, bn} /. {ToRules[r[n]]} //
> Simplify, {c[1], 0, 1}], 1], an | bn]; a[n_] := a[n] = Min[ab[n][[All,
> 1]]];
>
> Reap[ Do[ If[ Not[IntegerQ[Sqrt[n]] && Mod[n, 4] == 1], Print[{n, a[n]}];
> Sow[a[n]]], {n, 0, 69}]][[2, 1]]
>
> I get 3 more terms inserted  (352, 45, 5640) :
>
> data :
> {1, 1, 2, 1, 1, 8, 2, 10, 1, 40, 5, 2, 3, 250, 1, 1, 106, 3, 1138, 2,
>  8, 25, 146, 2968, 15, 298, 16, 2, 5, 17, 1856, 1, 1, 9384, 97, 10,
>  253970, 2, 72664, 3, 6440, 5, 521904, 1, 1, 3034, 5, 9148450,
>  1084152, 117, 2, 746, 10, 88, 157, 126890, 1, 1311, 56, 287};
>
> output from Mma :
> {1, 1, 2, 1, 1, 8, 2, 10, 1, 40, 5, 2, 3, 250, 1, 1, 106, 3, 1138, 2,
> 8, 25, 146, 1, 2968, 15, 298, 16, 2, 5, *352*, 17, 1856, 1, 1, 9384,
> 97, 10, 8, 253970, 2, 72664, 3, 6440, 5, 521904, 1, 3034, 5, 9148450,
> 3, 1084152, 117, 2, *45*, 746, 10, 88, 157, 126890, 1, 1, *5640*, 1311,
> 56, 287}
>
> Can some expert explain the trouble ?
>
> jfa
>
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