OK, the Mitchell/Smith/McCaughan theorem now shows when w=odd that X := reverse(X+1) (all mod 2^w) has period=2^w, and when w=even we get period=2*2^(w/2)-1. Empirically it seems like if w=odd then X := reverse(X+C), everything mod 2^w achieves period=2^w when the constant C is chosen randomly, with chance of the order of 2^(-0.49*w). The Theorem shows this chance is positive. Presumably the empirically fit value 0.49 really is 0.5? If w=even, then apparently period=2^w is unachievable even with the best choice of C; but by good choice of C, substantially greater period than by using C=1 is usually achievable (C and period in hexadecimal): w=2 C=3 period=3 w=4 C=9 period=B w=8 C=81 period=8F w=10 C=201 period=21F w=12 C=801 period=83F w=14 C=2001 period=207F w=16 C=8001 period=80FF and there seems to be a pattern here, doesn't there? Namely, when w=even and positive, C=1+2^(w-1) (which incidentally is palindromic) apparently causes period=2^(w-1)+2^(w/2)-1 which apparently is max possible. Meanwhile, when A and B are chosen randomly with A=odd, X := reverse(A*X+B), everything mod 2^w apparently achieves max period 2^w for each w>0, with positive chance of order 2^(-0.97*w), where I presume really 0.97 is 1? (We've only proven the chance positive when w=odd; but I also have example A,B for each w=1,2,3,...,20.)