PROBLEM NO.49 (B0MC-2,DAY-5)

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Let $p$ be a prime ,and let ${a_{k}}$ be an infinite sequence of integers for$k=0,1,2,......$ such that $a_{0}=0,a_{1}=1$,and $a_{k+2}=2a_{k+1}-pa_{k}$ for $k\ge 0$.if $-1$ appears in the sequence ,find all possible values of $p$.

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Hint:

Taking mod $p,p-1$ makes it easier.

[Phlembac Adib Hasan]

Hint:

Taking mod $p,p-1$ makes it easier.

No need to do so.$a_2=2.1-p.0=2$.
So $a_3=2.2-p.1=4-p$
Which gives if $p>5$, it's impossible to get $-1$ in the sequence as $a_i$ will be decreasing then.Now plug in $p=2,3,5$ which gives the result $p=5$.
Also replace the part $k\ge 0$ by $k\ge 2$ in the problem.

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many many thanks Adib.my solution took 2 pages.I used fermat's little theorem too.