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

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 \(\text{k} \leq 500\), except for \(\text{k} = 228, 282\) and \(444\), when \(\lambda = 3\). Using a machine, it is possible to do the checking for large \(k\).