On the existence of cyclic Hadamard difference sets

The main conjecture of this article is the following: if a cyclic (v=4n-1, k=2n-1, λ=n-1) Hadamard difference set exists, the the value of v must be either a prime, or a product of “twin primes,” or one less than a power of 2. Six cases, v=399, 495, 627, 651, 783, and 975, which were once listed as the possible exceptions for v<1000, are now fully investigated, and all the cases of v<10000 are now verified relative to this conjecture, with at most 17 possible exceptions