Trollathon #3.3 : The Mysterious Function
f ( 1 ) = 2 , f ( 2 ) = 4 , f ( 3 ) = 1 2 , f ( 4 ) = 2 6 2 1 4 8
Determine f ( 1 2 0 0 1 ) mod 2 0 1 4
The answer is 1932.
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1 solution
This is not necessarily correct. Consider f ( x ) = 4 3 6 8 7 x 3 − 2 6 2 1 1 9 x 2 + 4 8 0 5 5 0 x − 2 6 2 1 1 6 . Then, f ( 1 ) = 2 , f ( 2 ) = 4 , f ( 3 ) = 1 2 , f ( 4 ) = 2 6 2 1 4 8 , and f ( 1 2 0 0 1 ) = 7 5 4 7 2 2 6 4 6 9 6 4 7 6 0 0 2 , which is congruent to 1 2 7 8 ( m o d 2 0 1 4 ) .
Just note that f ( n ) = n f ( n − 1 ) − ( n − 1 ) + n and f ( 1 ) = 2 . and so by Euler's Theorem, we have 1 2 0 0 1 1 2 0 0 0 . . . 2 1 + 1 2 0 0 1 ≡ 1 2 0 0 2 ≡ 1 9 3 2 ( m o d 2 0 1 4 ) .