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for which n is a^n + 1/a^n integer if (a +1/a )^2=11 ?

Source: 15th -b QEDMO problem 12 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

May 30, 2021
algebraInteger

Problem Statement

Let aa be a real number such that (a+1a)2=11\left(a + \frac{1}{a}\right)^2=11. For which n∈Nn\in N is an+1ana^n + \frac{1}{a^n} an integer? Does this depend on the exact value of aa?
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