E453 -- Insignes proprietates serierum sub hoc termino generali contentarum \(x = \frac{1}{2}\!\big(a+\frac{b}{\sqrt{k}}\big)\left(p+q\sqrt{k}\right)^{n} + \frac{1}{2}\!\big(a-\frac{b}{\sqrt{k}}\big)\left(p-q\sqrt{k}\right)^{n}\)

(Eminent properties of series within which the general term is contained as \(x = \frac{1}{2}\!\big(a+\frac{b}{\sqrt{k}}\big)\left(p+q\sqrt{k}\right)^{n} + \frac{1}{2}\!\big(a-\frac{b}{\sqrt{k}}\big)\left(p-q\sqrt{k}\right)^{n}\))


Euler studies the \(n\)th order form \(fv^n + gu^n\). He gets a recursive relation of the form \(f(n+2)=2pf(n+1)-rf(n)\) and gets results on the numbers of the form \(p^2-kq^2\). Euler's work is made more difficult by the fact that he has not yet begun to use subscripts. In E453, he invents a notation [n] to denote the \(n\)th term of a sequence. Of course, he can only talk about one sequence at a time, but this notation is a significant improvement on anything he's used before.
This paper is best considered along with the number theory papers E452 and E454.

According to the records, it was presented to the St. Petersburg Academy on November 23, 1772.

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