Abstract: Introduction.In a recent investigation [7] the problem of factoring numbers of the form 22p + 1, p a prime, was encountered.Since 22p + 1 = (2P -2*<p+1) +1) (2P + 2è(p+1) + 1 ) for odd p, the problem consists of factoring the two trinomials on the right.In this paper the results of a search for factors of these trinomials are given, as well as a determination of the nature of certain of these numbers for which no factor was found.2. Elementary factors.Let Np = (2" -2i(p+I> + 1) (2" + 2è(p+1) + 1) = Ap ■Bp , p an odd prime.A. From the fact that 5 | Np , it easily follows that 5 | Ap iff p = ±1 (mod 8) and 5 | Bp iff p = ±3 (mod 8).On the other hand, 52 \ Np unless p = 5; for, since 2 is a primitive root of 25, 2 belongs to the exponent <j> (25) = 20.But 22p = -1 (mod 25), or 24p m 1 (mod 25).Therefore, 20 | 4p, or p = 5.Thus, if p = 5, 52 | 210 + 1 = 1025, while if p ^ 5, 52 \ Np .B. If g is a prime 5^5 and q \ Np , then 24p = 1 (mod q).But then 2 belongs to the exponent 4p (mod q).Thus by Fermat's Theorem, 4p | q -1 ; that is, every prime divisor 5^5 of Ap or Bp is =1 (mod 4p).C. Suppose p is odd and q = 4p + 1 is a prime.Then 2q~i = 24p = 1 (mod q).It follows from Euler's Criterion that 22p = (-) (mod q).But since p is odd, q = 5 (mod 8).Therefore, 22p = -1 (mod q), or q | 22p + 1.Unfortunately, however, it has not been possible to discover the conditions that determine which of Ap and Bp q will divide.3. The Search.A. Extent.The search for prime factors q ¿¿ 5 of AP and Bp , which was conducted on the IBM 701 at the University of California, Berkeley, was made over the following intervals:1 < q < y/Tfa for Bm 1 < q < 3-230 for An 1 < q < 230 for 71 < p ^ 179 and p = 241 1 < q < 228 for 179 < p < 1200, p * 241.No Np for p < 71, p t^ 59, were considered, since these numbers have been completely factored.A/24i was examined along with AV3 to the bound 230, these numbers being of particular interest (See [7]).B. Results, (i) The program produced a vast number of new factors, as well as several corrections to the literature (See [4]).The new factors of Np , p < 250, are indicated in the accompanying table by * to distinguish them from factors pre-