A078361 - OEIS

%I #12 Oct 03 2014 11:35:55

%S 1,3,8,5,5,46,12,64,7,7,302,39,16,25,2136,9,9,1000,29,11208,20,82,261,

%T 1552,11,11,33710,173,3488,190,24,61,4354,213,23550,13,13,124846,1305,

%U 136,110,3528264,28,1030190,43,93102,73,7688126,15,15,46312,77

%N Minimal positive solution a(n) of Pell equation a(n)^2 - D(n)*b(n)^2 = +4 or -4 with D(n)=A077425(n). The companion sequence is b(n)=A077058(n).

%C Computed from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values for the Diophantine eq. x^2 - x*y - ((D(n)-1)/4)*y^2= +1, resp., -1 if D(n)=A077425(n), resp, D(n)=A077425(n) and D(n) also in A077426 (this second case excludes in Perron's table the D values with a 'Teilnenner' in brackets).

%C The conversion from the x,y values of Perron's table to the minimal a=a(n) and b=b(n) solutions is a(n)=2*x(n)-y(n) and b(n)=y(n). If D(n)=A077425(n) is not in A077426 then the equation with -4 has no solution and a(n) and b(n) are the minimal solutions of the a(n)^2 - D(n)*b(n)^2 = +4 equation. If D(n)=A077425(n) is in A077426 then the a(n) and b(n) values are the minimal solution of the a(n)^2 - D(n)*b(n)^2 = -4 equation. In this case a(+,n)= a(n)^2+2 and b(+,n)=a(n)*b(n) are the minimal solution of a^2 - D(n)*b^2 = +4.

%C For Pell equation a^2 - D*b^2 = +4, see A077428 and A078355. For Pell equation a^2 - D*b^2 = -4, see A078356 and A078357.

%D O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

%e 29=D(5)=A077425(5) is A077426(4), hence a(5)=5 and b(5)=A077058(5)=1 solve a^2 - 29*b^2=-4 minimally and a(+,5)=a(5)^2+2=27 with b(+,5)=a(5)*b(5)=5*1=5 solve a^2 - 29*b^2=+4 minimally. See also A077428 with companion A078355.

%e 21=D(4)=A077425(4) is not in A077426, hence a(4)=5 and b(4)=A077058(4)=1 give the solution with minimal positive b of a^2 - 21*b^2=+4.

%K nonn

%O 1,2

%A _Wolfdieter Lang_, Nov 29 2002

%E More terms from _Matthew Conroy_, Apr 20 2003