BLOCK DESIGNS, PERMUTATION GROUPS AND PRIME VALUES OF POLYNOMIALS

A recent construction by Amarra, Devillers and Praeger of block designs with specific parameters and large symmetry groups depends on certain quadratic polynomials, with integer coefficients, taking prime power values. Similarly, a recent construction by Hujdurovic & PRIME;, Kutnar, Kuzma, Marus ˇic ˇ, Miklavic ˇ and Orel of permutation groups with specific intersection densities depends on certain cyclotomic polynomials taking prime values. The Bunyakovsky Conjecture, if true, would imply that each of these polynomials takes infinitely many prime values, giving infinite families of block designs and permutation groups with the required properties. We have found large numbers of prime values of these polynomials, and the numbers found agree very closely with the estimates for them provided by Li's recent modification of the Bateman-Horn Conjecture. While this does not prove that these polynomials take infinitely many prime values, it provides strong evidence for this, and it also adds extra support for the validity of the Bunyakovsky and Bateman-Horn Conjectures.