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单词 ExclusionOfIntegerRoot
释义

exclusion of integer root


Theorem.  The equation

p(x):=anxn+an-1xn-1++a0= 0

with integer coefficients ai has no integer roots (http://planetmath.org/Equation), if p(0) and p(1) are odd.

Proof.  Make the antithesis, that there is an integer x0 such that  p(x0)=0.  This x0 cannot be even, because else all terms of p(x0) except a0 were even and thus the whole sum could not have the even value 0.  Consequently, x0 and also its powers (http://planetmath.org/GeneralAssociativity) have to be odd.  Since

20=p(x0)and2p(0)=a0,

there must be among the coefficients an,an-1,,a1 an odd amount of odd numbersMathworldPlanetmathPlanetmath.  This means that

2an+an-1++a1+a0=p(1).

This however contradicts the assumptionPlanetmathPlanetmath on the parity of p(1), whence the antithesis is wrong and the theorem .

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更新时间:2025/5/4 22:03:39