To supplement existing data, solutions of a p−1 ≡ 1 (mod p 2 ) are tabulated for primes a, p with 100 < a < 1000 and 10 4 < p < 10 11 . For a < 100, five new solutions p > 2 32 are presented. One of these, p = 188748146801 for a = 5, also satisfies the "reverse" congruence p a−1 ≡ 1 (mod a 2 ). An effective procedure for searching for such "double solutions" is described and applied to the range a < 10 6 , p < max (10 11 , a 2 ). Previous to this, congruences a p−1 ≡ 1 (mod p r ) are generally considered for any r ≥ 2 and fixed prime p to see where the smallest prime solution a occurs.

Abstract. Extending previous searches for prime Fibonacci and Lucas num-

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Abstract.Numbers of the forms C"=«-2" + l and W" = n-2" -\ are both called Cullen numbers. New primes C" are presented for n = 4713 , 5795 , 6611 , 18496 . For Wn , several new primes are listed, the largest one having n = 18885 . Furthermore, all efforts made to factorize numbers Cn and W" are described, and the result, the complete factorization for all n < 300, is given in a Supplement.

Abstract.Generalized Fermât numbers have the form Fb m. It is shown that each prime is a factor of some F¡, m for approximately 1 /k bases b , independent of n . Divisors of generalized Fermât numbers of base 6, base 10, and base 12 are tabulated. Three new factors of standard Fermât numbers are included.