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7

Part of PEN L Problems

Problems(1)

L 7

Source:

5/25/2007
Let mm be a positive integer. Define the sequence {an}n≥0\{a_{n}\}_{n \ge 0} by a0=0,  a1=m,  an+1=m2an−an−1.a_{0}=0, \; a_{1}=m, \; a_{n+1}=m^{2}a_{n}-a_{n-1}. Prove that an ordered pair (a,b)(a, b) of non-negative integers, with a≤ba \le b, gives a solution to the equation a2+b2ab+1=m2\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 \ge 0.
Linear Recurrencespen