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Omni-MATH / Find the smallest positive integer n such that the polynomial (x+1)^(n)−1 is "divisible by x²+1 modulo 3", or more precisely, either of the following equivalent conditions holds: t…

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problem

Find the smallest positive integer nn such that the polynomial (x+1)n−1(x+1)^{n}-1 is "divisible by x2+1x^{2}+1 modulo 3", or more precisely, either of the following equivalent conditions holds: there exist polynomials P,QP, Q with integer coefficients such that (x+1)n−1=(x2+1)P(x)+3Q(x)(x+1)^{n}-1=\left(x^{2}+1\right) P(x)+3 Q(x); or more conceptually, the remainder when (the polynomial) (x+1)n−1(x+1)^{n}-1 is divided by (the polynomial) x2+1x^{2}+1 is a polynomial with (integer) coefficients all divisible by 3.
Plain-text mathematical notation (without MathML)
Find the smallest positive integer n such that the polynomial (x+1)^(n)−1 is "divisible by x²+1 modulo 3", or more precisely, either of the following equivalent conditions holds: there exist polynomials P,Q with integer coefficients such that (x+1)^(n)−1=(x²+1)P(x)+3Q(x); or more conceptually, the remainder when (the polynomial) (x+1)^(n)−1 is divided by (the polynomial) x²+1 is a polynomial with (integer) coefficients all divisible by 3.
Original LaTeX notation
Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ is "divisible by $x^{2}+1$ modulo 3", or more precisely, either of the following equivalent conditions holds: there exist polynomials $P, Q$ with integer coefficients such that $(x+1)^{n}-1=\left(x^{2}+1\right) P(x)+3 Q(x)$; or more conceptually, the remainder when (the polynomial) $(x+1)^{n}-1$ is divided by (the polynomial) $x^{2}+1$ is a polynomial with (integer) coefficients all divisible by 3.

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Source and history

task_id
f02645e8-d7ab-5cfa-b1ee-2dfb4b23b83d
task_key
test--f02645e8-d7ab-5cfa-b1ee-2dfb4b23b83d
task_revision_id
1
config / split
/ test

Official source

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