My Ph.D. thesis (co-written with Marc-Paul van der Hulst, under supervision of H.W. Lenstra, Jr., P. van Emde Boas, and A.K. Lenstra) concerns improvements to the general primality proving method using Jacobi sums, to which the names of Adleman, Pomerance, Rumely, Cohen and Lenstra are usually attached.
For numbers of the form h*2^k+1 and h*2^k-1 special, very efficient tests have been developed. These generalize the well-known tests for Fermat primes and Mersenne primes, the cases with h=1, used to find the largest known primes. In the paper mentioned below I consider the question of finding finitely many `starting values' for these tests which can be used for all k with 2^k > h, if h is fixed (the case with h divisible by 3 is the main problem). Explicit starting values were found (using Magma) for all cases with h less than 100000, with the exception of those h that are of the form 2^m - 1, for which it is proved that a finite solution in the above sense does not exist.
In the `scriptie' for my `kandidaatsexamen' I considered certain analogues of the primality tests of Lucas-Lehmer type as described above using elliptic curves with complex multiplication.