This “Week’s” Puzzle
Even though the cubic function
f(x) = x3-x2-3x+5
is non-invertible over real numbers, when it is applied to the integers modulo
10 it is invertible. Thus it is said to be a permutation modulo
10. For each integer y in the range 0..9 there is exactly one x for which
f(x) = y, as can be seen from the table below.
| x |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| x3-x2-3x+5 mod 10 |
5 | 2 | 3 | 4 | 1 | 0 | 7 | 8 | 9 | 6 | |
Show that if p is an odd prime, no quadratic
g(x) = ax2+bx+c
is a permutation modulo p.
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