Numerical experiments on a conjecture of B.C. Mortimer and K.S. Williams
Masahiko Sato, Masataka Yorinaga · Proceedings of the Japan Academy Series A Mathematical Sciences · 1973
Let p be a rational prime and n a positive integer _>_2.We denote by a(p) the least positive integral value of a for which the polynomial x + x + a is irreducible (mod p), and set a-lim inf a(p).B. C. Mortimer and K. S. Williams [2] have stated the following Conjecture.Put a* --1 and for n >= 3 define l if n--0, 1 (mod 3), a* 2 if n=2 (mod6), 3 if n-5 (mod6).Then we have an=a*.K. S. Williams [5] proved that this conjecture is in fact true for n=2 and 3, and Mortimer and Williams [2] verified the conjecture for all n__<20 with the aid of a computer.The results of S. Uchiyama [4] show that the conjecture is true whenever n itself is a prime number.In 1 of the present paper we shall show that the conjecture is.true for all n=< 40 by making use of an algorithm which is faster than the one used in [2].As to the discriminant D of the polynomial x+x+a*, it is possible to examine the values of it for a fairly wider range of n, and we observe in 2 some arithmetical properties of D that will be of an independent interest.The computations in 1 were accomplished by the first-named author and those in 2 were done by the second-named author.