On primitive trinomials (Mod 2)

摘要: The present paper is a completion of previous the same title (Zierler and Brillhart, 1968). In our preceding work 187 irreducible trinomials T~.k(x) = x" ~ x k ~- 1 were left to be tested for primitivity at later date, even though requisite complete factorizations 2 - known (these identified in 1968) by superscript minus sign on n). This testing has now been done CDC 6600 Communications Research Division Institute Defense Analyses, Princeton (the Zierler Brillhart (1968) was both this computer IBM 7094 Bell Telephone Laboratories, ttolmdel, New Jersey). results are given Table 1. italic entries table refer primitive trinomials, while those that not italicized whose periods less than For imprimitive Tn,~(x) with (n, k) we have index (= (2 1)/Period) rather period T,~(x), since extremely large. remaining T. .k(x) > 1, entirely omitted giving their periods, because ease which these can calculated from Tables information Relevant calculation following theorem (Berlekamp, 1968), p. 153: THEOaEM: Let f(x) an polynomial n over GF (p~), let t~ p pmme. If t [ n, then every factor f(x ~) tn. one other factors