Lucas-Lehmer Test With Polynomials

[Milkman]

Does anyone know why the lucas-lehmer test won't work with numbers represented by polynomials? The modulus polynomial does not divide the last term correctly for some reason.
Example (the code boxes are polynomial long division):

the numbers will be represented in base 4, so 4 = x

p = x + 1 (or 5; the mersenne exponent)
2^p-1 = x^2 + 3x + 3 (or 31; the mersenne number to be tested)

p-2 iterations of the lucas sequence, mod 2^p-1:

(x)^2 - 2 = x^2 - 2
x^2 - 2 = -3x - 5 (mod x^2 + 3x + 3)

               1
             ---------------
x^2 + 3x + 3 ) x^2 + 0x - 2
              -x^2 - 3x - 3
              --------------
                    -3x - 5

(-3x - 5)^2 - 2 = 9x^2 + 30x + 23
9x^2 + 30x + 23 = 3x - 4 (mod x^2 + 3x + 3)

                9
             ------------------
x^2 + 3x + 3 ) 9x^2 + 30x + 23
              -9x^2 - 27x - 27
              ----------------
                       3x - 4

(3x - 4)^2 - 2 = 9x^2 - 24x + 14
9x^2 - 24x + 14 = -51x - 13 (mod x^2 + 3x + 3)

               9
             ------------------
x^2 + 3x + 3 ) 9x^2 - 24x + 14
              -9x^2 - 27x - 27
              ----------------
                     -51x - 13

x^2 + 3x + 3 should divide 9x^2 - 24x + 14 evenly, and the last result should be 0, not -51x - 13. And -51x - 13 is not congruent to zero in any field (I don't think).
So what's wrong?

 

[T.Rex]

Don't we should have all coefficients lower than x=4 ?
As an example, 3x - 4 = 2x ?
T.

 

[Hurkyl]

x^2 + 3x + 3 should divide 9x^2 - 24x + 14 evenly

That's not what Lucas-Lehmer says: it says that

4² + 3*4 + 3

should divide

9*4² - 24*4 + 14

evenly. (And it does)