#### English Title

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}

#### Enestrom Number

453

#### Fuss Index

146

#### Original Language

Latin

#### Content Summary

Euler studies the *n*th order form *fv ^{n}*+

*gu*. He derives the recursive relation

^{n}*f*(

*n*+2) = 2

*p∙f*(

*n*+1) −

*r∙f*(

*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. Instead, he invents a notation [

*n*] to denote the

*n*th 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.

#### Published as

Journal article

#### Published Date

1774

#### Written Date

1772

#### Original Source Citation

Novi Commentarii academiae scientiarum Petropolitanae, Volume 18, pp. 198-217.

#### Opera Omnia Citation

Series 1, Volume 15, pp.185-206.

#### Record Created

2018-09-25