Let P and Q be polynomials with integer coefficients such that P(a) divides Q(a) in Z for every integer a. Does it mean that P divides Q in Z[X]?P(a) divides Q(a) means that there exists an integer k(a) such that Q ( a ) = k ( a ) P ( a ), so Q = k P. This means that P divides Q if k is polynomial with integer coefficients. But I can't find a proof or a counter-example. Any idea?For the context, I was looking at a proof of the fact that X 2 − X + 1 divides X 2 n + 1 + ( X = 1 ) n + 2 for every n ≥ 1 (in a video of Michael Penn I think but I am not sure). The method used was to consider congruences modulo P ( X ) = X 2 − X + 1. This is fine and clever method, but my students don't know arithmetic in Z[X], so I tried to circumvent the problem by looking at congruence in Z modulo P(a) for every a ∈ Z . But I hit the problem stated above to conclude the reasoning.

Question

Let P and Q be polynomials with integer coefficients such that P(a) divides Q(a) in Z for every integer a. Does it mean that P divides Q in Z[X]?P(a) divides Q(a) means that there exists an integer k(a) such that   Q  (  a  )  =  k  (  a  )  P  (  a  ), so   Q  =  k  P. This means that P divides Q if k is polynomial with integer coefficients. But I can&#039;t find a proof or a counter-example. Any idea?For the context, I was looking at a proof of the fact that       X    2    −  X  +  1 divides       X          2      n      +      1        +  (  X  =  1      )          n      +      2       for every   n  ≥  1 (in a video of Michael Penn I think but I am not sure). The method used was to consider congruences modulo   P  (  X  )  =      X    2    −  X  +  1. This is fine and clever method, but my students don&#039;t know arithmetic in Z[X], so I tried to circumvent the problem by looking at congruence in Z modulo P(a) for every   a  ∈      Z  . But I hit the problem stated above to conclude the reasoning.

teleriasacr · Accepted Answer

Step 1Does it mean that P divides Q in Z[X]No, take for example     P  (  X  )  =  2  ,    Q  (  X  )  =      X    2    +  X  .Step 2      X    2    −  X  +  1 divides       X          2      n      +      1        +  (  X  −  1      )          n      +      2      If         x    2    −  x  +  1  =  0   then     x  −  1  =      x    2    , so         x          2      n      +      1        +  (  x  −  1      )          n      +      2        =      x          2      n      +      1        +      x          2      n      +      4        =  …

Let P and Q be polynomials with integer coefficients such that P(a) divides Q(a) in Z for every integer a. Does it mean that P divides Q in Z[X]?

Answered question

Answer & Explanation

New Questions in Algebra II