OFFSET
1,1
COMMENTS
Jacobson & Williams proved that rho(n) can be arbitrarily large, therefore this sequence is infinite.
Of the first 40 terms only 6 is composite.
LINKS
Michael J. Jacobson and Hugh C. Williams, The size of the fundamental solutions of consecutive Pell equations, Experimental Mathematics, Vol. 9, No. 4 (2000), pp. 631-640. alternative link.
EXAMPLE
n = 61 is in the sequence since the least positive solution to x^2-60*y^2 = 1 has x = 31, and the least positive solution to x^2-61*y^2 = 1 has x = 1766319049, so rho(61) = log(1766319049)/log(31) = 6.200... larger than for any smaller n.
MATHEMATICA
$MaxExtraPrecision= 1000; a[n_]:=If[IntegerQ[Sqrt[n]], 0, For[y=1, !IntegerQ[ Sqrt[n*y^2+1]], y++, Null]; y]; PellSolve[(m_Integer)?Positive] := Module[ {cf, n, s}, cof = ContinuedFraction[Sqrt[m]]; n = Length[ Last[cof]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; f[n_] := If[ !IntegerQ[ Sqrt[n]], PellSolve[n][[1]], 0]; rho[x0_, x1_]:=If[x0==0||x1==0, 0, Log[x1]/Log[x0]]; x0=2; n=3; rhom=0; seq={}; Do[x1=f[n]; rho1 = rho[x0, x1]; If[rho1 > rhom, AppendTo[seq, n]; rhom=rho1]; x0=x1; n++, {k, 1, 1000}]; seq
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 26 2018
STATUS
approved