Just a curiosity, but two of the six exceptions below fit into the (q-1)=k(p-1) test, albeit with composite p. 151*751*28351 = 28351 * 113401 151*2551*192601 = 192601 * 385201 (hard #1) --Rich

--actually, even more: 331*2971*49171 = 49171*983401, note 983400/49170=20 (hard #2) where note the only 2 "hard" ones among Jaeschke's 6 exceptions are covered by the (q-1)=k*(p-1) test if k=20 is also considered. --Warren D. Smith