MathDB
Equation and sequence

Source: Canada 1998

March 4, 2006
number theory unsolvednumber theory

Problem Statement

Let mm be a positive integer. Define the sequence a0,a1,a2,⋯a_0, a_1, a_2, \cdots by a0=0,  a1=m,a_0 = 0,\; a_1 = m, and an+1=m2an−an−1a_{n+1} = m^2a_n - a_{n-1} for n=1,2,3,⋯n = 1,2,3,\cdots. Prove that an ordered pair (a,b)(a,b) of non-negative integers, with a≤ba \leq b, gives a solution to the equation a2+b2ab+1=m2 {\displaystyle \frac{a^2 + b^2}{ab + 1} = m^2} if and only if (a,b)(a,b) is of the form (an,an+1)(a_n,a_{n+1}) for some n≥0n \geq 0.
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