MathDB

Let

\alpha \leq -2

be an integer. Prove that for every pair

(\beta_0, \beta_1)

of integers there exists a uniquely determined sequence

0\leq q_0, \ldots, q_k<\alpha ^ 2 - \alpha

of integers, such that

q_k\neq 0

(\beta_0, \beta 1)\neq (0,0)

and

\beta_i=\sum_{j=0}^k q_j(\alpha - i)^j,\text{ for }i=0,1

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