1969 IMO Problems/Problem 1

Problem

Prove that there are infinitely many natural numbers $a$ with the following property: the number $z = n^4 + a$ is not prime for any natural number $n$.

Solution

Suppose that $a = 4k^4$ for some $a$. We will prove that $a$ satisfies the property outlined above.

The polynomial $n^4 + 4k^4$ can be factored as follows:

\[n^4 + 4k^4\] \[= n^4 + 4n^2k^2 + 4k^4 - 4n^2k^2\] \[= (n^2 + 2k^2)^2 - (2nk)^2\] \[= (n^2 + 2k^2 - 2nk)(n^2 + 2k^2 + 2nk)\]

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 $n^2 + 2k^2 - 2nk > 1$ when $k > 1$. Thus, for all $k > 2$, $4k^4$ is a valid value of $a$, completing the proof. $\square$

~mathboy100

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

See Also

1969 IMO (Problems) • Resources
Preceded by
First question
1 2 3 4 5 6 Followed by
Problem 2
All IMO Problems and Solutions
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