The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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Heilermann's' method has an interest in that he brings
out the two Archimedian approximations in immediate
sequence without intervening values. He generalizes the
side- and diagonal numbers of Theon of Smyrna, which
furnished solutions for the equations
X2 - 2y2 = - 1,
into solutions of the general Pell equation
X2 - ay2 = b.
From Theon we might have
Sn = Sn-1 + D_, Dn n = 2Sn- + Dn-1.
Heilermann set
S1 = So + Do) D1 = aSo + Do,
S2 = Si + D1, D2 = aSD + D1,
S3 = S2 +, D3 = aS2 + D2,
*..................
Sn = Sn-_ + Dn-1, Dn = aSn-1 + Dn1.
Therefore
aS,2 = aS,-_2 + 2aSn_lDn_l + aDn-2,
Dn2 = a2Sn-2 + 2aSn-_Dn-_ + Dn1_2.
Subtracting,
Dn2 - aSn2 = (1 - a)(D,_2 - aSn_2)
= (1 - a)2(Dn_22 - aS_22)
= (1 - a)n(D2- aS2);
and, if
Do = So = 1,
Dn2 - aSn2 = (1 -a),+l;
and since
D,n + (1 - a)"n+
S n- (1 S-2
1 Heilermann, "Bemerkungen zu den Archimedischen Naherungswerten
der irrationalen Quadratwurzeln," Zeitschrift fiir Mathematik und Physik,
Historisch-literarische Abtheilung, vol. XXVI, p. 121, Leipzig, 1881.
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