1969 IMO Problems/Problem 1
Prove that there are infinitely many natural numbers with the following property: the number is not prime for any natural number .
Suppose that for some . We will prove that satisfies the property outlined above.
The polynomial can be factored as follows:
Both factors are positive, because if the left one is negative, then the right one would also negative, which is clearly false.
It is also simple to prove that when . Thus, for all , is a valid value of , completing the proof.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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