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

The equation was $z = n^4 + a$ ,you can put $a = 4 m^4$ for all natural numbers m. So you will get $z = n^4 + 4 m^4 = n^4+4m^4 +4n^2 m^2 - 4n^2 m^2$ $z = (n^2+2 m^2)^2 - (2nm)^2 = (n^2+2 m^2 -2nm)(n^2+2 m^2 +2nm)$ so you get $z$ is composite for all $a = 4 m^4$

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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