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a_n

Source: Iran 2005

August 29, 2005
number theory proposednumber theory

Problem Statement

kk is an integer. We define the sequence {an}n=0∞\{a_n\}_{n=0}^{\infty} like this: a0=0,   a1=1,   an=2kan−1−(k2+1)an−2  (n≥2)a_0=0,\ \ \ a_1=1,\ \ \ a_n=2ka_{n-1}-(k^2+1)a_{n-2}\ \ (n \geq 2) pp is a prime number that p\equiv 3(\mbox{mod}\ 4) a) Prove that a_{n+p^2-1}\equiv a_n(\mbox{mod}\ p) b) Prove that a_{n+p^3-p}\equiv a_n(\mbox{mod}\ p^2)
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