Thread: mod x^4+1 View Single Post 2022-03-23, 00:07   #26
Dr Sardonicus

Feb 2017
Nowhere

7·857 Posts Quote:
 Originally Posted by paulunderwood I don't see why it has to be base 2 Fermat PRP; The norm is -1.
By "your condition" I mean, any of Mod(Mod(1,n) + x, x^4 + 1)^n == Mod(1 + x^r, x^4 + 1) for n == r (mod 8), r =3, 5, or 7.

Any of these conditions imply that Mod(2,n)^n = Mod(2,n).

My "reduced" condition does NOT imply that Mod(2,n)^n = Mod(2,n) (see example next paragraph). It only covers part of the test for n == 5 (mod 8). It merely insures that in Mod(Mod(1,n) + x, x^4 + 1)^n, the coefficients of x^3 and x^2 are 0.

A mindless numerical sweep found that 1189 = 29*41 is not a 2-psp but does satisfy my "reduced" condition.

I note that 341, a 2-psp congruent to 5 (mod 8), does not satisfy my "reduced" condition.

So combining 2-psp with my "reduced" condition may work fairly well for n == 5 (mod 8), but I suspect there are 2-psp's congruent to 5 (mod 8) which satisfy my "reduced" condition, but not your condition

Mod(Mod(1,n) + x, x^4 + 1) = Mod(Mod(1,n) - x, x^4 + 1).  