Understand the problem
Source of the problem
Start with hints
The sequence cannot be decreasing because it is a sequence of positive integers. Hence there exists (a smallest) such that . If then the sequence becomes constant from the th term onwards (we shall treat this case later). Otherwise hence . This implies that the sequence becomes increasing from the th term onwards. Also, for . This difference equation has the characteristic equation (see the link in hint 1) which has the solutions . Thus, for satisfying . Take any prime divisor of . By Fermat’s little theorem, for every positive integer . Thus . Hence the sequence contains infinitely many composites. This cannot be allowed, so the sequence cannot be strictly increasing at any point.
The above discussion shows that, for any permissible sequence, there exists a (smallest) and a prime such that for all . For , the sequence is decreasing. Note that, either or . Hence, either or . The first one can happen only if because otherwise the minimality of is violated. In that case, the sequence is constant and . If then either (in which case ) or hence . The last equality forces to be 2. Thus . If then which is absurd as cannot be an odd number. Hence in this case and .
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