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This calculation makes the kernel crash because it needs so much memory. Thoughts on how to get around this?

PolynomialRemainder[1+x^267910657,1+x+x^20+x^32, x, Modulus->2]
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PolynomialMod[1 + x^267910657, {1 + x + x^20 + x^32, 2}] – whuber May 1 '13 at 19:09
I have to ask, where did this come from? It is not something I would expect to do, so I'm curious. – rcollyer May 1 '13 at 20:09
We'll look into improving on this. – Daniel Lichtblau May 1 '13 at 22:05
I get an error message CoefficientList::lrgexp: Exponent is out of bounds for function CoefficientList. >> (v9.0.1 Mac). – Michael E2 May 2 '13 at 0:48

I think, "Fermat little theorem" can work on your problem. Also, you can find it here

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Could you amplify this answer and show how your suggestion would work in Mathematica? It's especially puzzling that the reference you give makes no mention at all of polynomials. – whuber May 1 '13 at 21:23
Taking @whuber's comment a step further, as polynomials, x^p and x are by no means equivalent modulo p. – Daniel Lichtblau May 1 '13 at 22:04
Yes. you are right. if katie tries to find some solution, then it runs in my opinion. Because Fermat Little Theorem reduce the exponent of x. So, the theorem makes it easy. Unfortunately, we don't know the main problem. – MATIRMAK May 1 '13 at 22:16

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