Ternary cyclotomic polynomials having a large coefficient

Let ${\mathrm{\Phi }}_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in ${\mathrm{\Phi }}_n(x)$, satisfies $\mathrm{\mid }{a}_n(k)\mathrm{\mid }\le (p+1)\mathrm{/}2$ in case $n=pqr$ with $p<q<r$ primes (in this case ${\mathrm{\Phi }}_n(x)$ is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example $\mathrm{\mid }{a}_n(k)\mathrm{\mid }\le 3p\mathrm{/}4$). Here we show that, nevertheless, Beiter’s conjecture is false for every $p\ge 11$\mathrm{.}Wealsoprovethatgivenany$\epsilon>0$ there exist infinitely many triples $(p_j,q_j,r_j)$ with $p_1<p_2<\mathrm{.}\mathrm{.}\mathrm{.}$ consecutive primes such that $\mathrm{\mid }{a}_{p_j{q}_j{r}_j}(n_j)\mathrm{\mid }>(2\mathrm{/}3-ϵ)p_j$ for $j\ge 1$.