Insignes proprietates serierum sub hoc termino generali contentarum x = 1/2∙(a+b/√k)(p+q√k)n + 1/2∙(a-b/√k)(p-q√k)n
Eminent properties of series within which the general term is contained as x = 1/2∙(a+b/√k)(p+q√k)n + 1/2∙(a-b/√k)(p-q√k)n
Euler studies the nth order form fvn+gun. He derives the recursive relation f(n+2) = 2p∙f(n+1) − r∙f(n) and gets results on the numbers of the form p2−kq2. Euler's work is made more difficult by the fact that he has not yet begun to use subscripts. Instead, he invents a notation [n] to denote the nth term of a sequence. Of course, he can only refer one sequence at a time with this notation, but it is a significant improvement on anything he's used before. This paper is best considered alongside the number theory papers E452 and E454.
Original Source Citation
Novi Commentarii academiae scientiarum Petropolitanae, Volume 18, pp. 198-217.
Opera Omnia Citation
Series 1, Volume 15, pp.185-206.