Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
Already signed up?
New to KöMaL?

Problem B. 5215. (January 2022)

B. 5215. Find all positive real numbers \(\displaystyle x\) for which \(\displaystyle x + \frac1{x}\) is an integer, and \(\displaystyle x^3 + \frac1{x^3}\) is a prime number.

Based on the idea of B. and V. Szaszkó-Bogár

(4 pont)

Deadline expired on February 10, 2022.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Ha \(\displaystyle x + \frac1{x}=n\) pozitív egész, akkor négyzetre emelve \(\displaystyle x^2 + \frac1{x^2} + 2=n^2\), azaz \(\displaystyle x^2 + \frac1{x^2}=n^2-2\) is pozitív egész, ezért \(\displaystyle n>1\). A két egyenletet összeszorozva:

\(\displaystyle x^3 + \frac1{x^3} + x + \frac1{x} = n^3-2n, \)

vagyis \(\displaystyle x^3 + \frac1{x^3} = n^3-3n = n(n^2-3)\) két pozitív egész szorzata. Ez pontosan akkor prímszám, ha az egyik tényező 1, a másik pedig prím. Mivel \(\displaystyle n>1\), azért \(\displaystyle n^2-3=1\), és így \(\displaystyle n=2\). Tehát \(\displaystyle x + \frac1{x}=2\) alapján \(\displaystyle x=1\).


Statistics:

142 students sent a solution.
4 points:125 students.
3 points:7 students.
2 points:1 student.
1 point:2 students.
0 point:4 students.
Unfair, not evaluated:1 solutions.

Problems in Mathematics of KöMaL, January 2022