Arithmetic Sequences and Second order Recurrences

Abstract

In [1], M. K. Mahanthappa considers the three equations

$$ n{x^2} + \left( {n + 1} \right)x - \left( {n + 2} \right) = 0 $$

(1)

$$ n{x^2} + \left( {n + 2} \right)x - \left( {n + 1} \right) = 0 $$

(2)

$$ n{x^2} + \left( {n - 1} \right)x - \left( {n + 1} \right) = 0 $$

(3)

where, apart from sign, the three coefficients are integers in arithmetic progression with common difference 1 and n positive. He asks when each equation has rational solutions and concludes that this is so in (1) if and only if n = F_{2m + 1} − 1 when the solutions are F _2m /(F _{2m + 1} − 1) and − F _{2m + 2}/(F _{2m + 1} − 1), m a positive integer; in (2) if and only if n = F _{2m + 3}F_2m where the solutions are F _{2m + 2}/F _{2m + 3} and − F _{2m + 1}/F _2m , m a positive integer; and in (3) if and only if n = F _{2m + 1} F _2m when the solutions are F _{2m − 1}/F _2m and − F _{2m + 2}/F _{2m + 1}, m a positive integer. Of course, as is usual in this journal, F _n denotes the nth Fibonacci number. The more general case of equations ax ² + bx − c = 0 with a, b, and c integers in arithmetic progression with common difference r was suggested by Mahanthappa but not treated. We consider the more general case here.