Difference between revisions of "2008 iTest Problems/Problem 100"
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| − | + | Let <math>\alpha</math> be a root of <math>x^6-x-1</math>, and call two polynomials <math>p</math> and <math>q</math> with integer coefficients <math>\textit{equivalent}</math> if <math>p(\alpha)\equiv q(\alpha)\pmod3</math>. It is known that every such polynomial is equivalent to exactly one of <math>0,1,x,x^2,\ldots,x^{727}</math>. Find the largest integer <math>n<728</math> for which there exists a polynomial <math>p</math> such that <math>p^3-p-x^n</math> is equivalent to <math>0</math>. | |
Latest revision as of 15:15, 15 August 2018
Let 
be a root of 
, and call two polynomials 
and 
with integer coefficients 
if 
. It is known that every such polynomial is equivalent to exactly one of 
. Find the largest integer 
for which there exists a polynomial such that 
is equivalent to 
.
