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recurrence relation, parity of a_{n+1}a_n

Source: VI Soros Olympiad 1990-00 R3 9.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 28, 2024
algebranumber theory

Problem Statement

The sequence of integers a1,a2,a3,..a_1,a_2,a_3 ,.. such that a1=1a_1 = 1, a2=2a_2 = 2 and for every natural n≥1n \ge 1
an+2={2001an+1−1999an,  if  the  product  an+1anis  an  even  number/an+1−an,  if  the  product  an+1anis  an  odd  numbera_{n+2}=\begin{cases} 2001a_{n+1} - 1999a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,even\,\,number} /\\ a_{n+1}-a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,odd\,\,number} \end{cases}
Is there such a natural mm that am=2000a_m= 2000?
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