In this paper, criteria of divisibility of the class number h + of the real cyclotomic field Q(ζp + ζ −1 p ) of a prime conductor p and of a prime degree l by primes q the order modulo l of which is l−1 2 , are given. A corollary of these criteria is the possibility to make a computational proof that a given q does not divide h + for any p (conductor) such that both p−1 2 , p−3 4 are primes. Note that on the basis of Schinzel's hypothesis there are infinitely many such primes p.