## Sur un système linéaire de Terjanian

### Volume 182 / 2018

Acta Arithmetica 182 (2018), 331-345
MSC: Primary 11D04; Secondary 11D41, 11A15.
DOI: 10.4064/aa170331-10-11
Published online: 22 January 2018

#### Abstract

In 1989, Terjanian tried to prove the first case of Fermat’s Last Theorem for any odd prime exponent $l$ using cyclotomic units and Hilbert symbols. His proof was incomplete since it relied on a conjecture, named $LC$. He also defined a property $P$ that implies $LC$ and that is phrased simply using linear equations over the finite field with $l$ elements. We prove that this property $P$ is false for almost all prime numbers $l$, and in particular when $l-1$ has a prime factor greater than or equal to $11$.