Validity of Lander’s Conjecture for $\lambda$ = 3 and $k\le 500$

Abstract

Lander Conjectured: If $D$ is a $(\text{v, k},\lambda )$ difference set in an abelian group G with a cyclic Sylow p-subgroup, then p does not divide $(v,n)$, where $\text{n}=\text{k}–\lambda$.

In a previous paper, the above conjecture was verified for $\lambda =3$ and $\text{k}\le 500$, except for $\text{k}=228,282$ and $444$. These three exceptional values are dealt with in this note, thereby verifying Lander’s conjecture completely for $\lambda =3$ and $\text{k}\le 500$.