On the congruences a^n +/- b^n = 0 mod n^k

Salvatore Tringali (1)

1 LJLL - Laboratoire Jacques-Louis Lions

We determine all triples $(a,b,n)$ of integers with $\gcd(a,b) = 1$ and $n \ge 1$ such that $n^k$ divides $a^n + b^n$ for $k = \max(|a|,|b|)$. In particular, for positive integers $m,n$ we show that $n^m \mid m^n + 1$ if and only if either $(m,n)=(2,3)$, $(m,n) = (1,2)$, or $n=1$ and $m$ is arbitrary; this generalizes a couple of problems from the 1990 and 1999 editions of the International Mathematical Olympiad. Then we solve the same question with $a^n - b^n$ in place of $a^n + b^n$. The results are related to a conjecture by K. Gy\H{o}ry and C. Smyth on the finiteness of $\{n \in \mathbb N^+: n^k \mid a^n \pm b^n\}$ when $a,b,k$ are fixed integers with $k \ge 3$, $\gcd(a,b) = 1$, and $|a|,|b|$ not simultaneously equal to $1$.

Mots clés: Congruences; exponential Diophantine equations

Domaines: Théorie des nombres [math.NT]

Salvatore Tringali. On the congruences a^n +/- b^n = 0 mod n^k. 2013. ⟨hal-00796514⟩