On Lander’s Conjecture for the Case $\lambda =3$

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 $\text{n = k}-\lambda$.

Various nonexistence theorems are used to verify the above conjecture (all hand calculations) for $\text{k}\le 500$, except for $\text{k}=228,282$ and $444$, when $\lambda =3$. Using a machine, it is possible to do the checking for large $k$.