Let $P(x)$ be a polynomial with integer coefficients. Show that there is a non-zero polynomial $Q(x)$ with integer coefficients, such that the product $$P(x)Q(x)=\sum_{k\ge 0}a_k x^k$$ has only nonzero coefficients at the prime degree terms. In other words, $a_k=0$ if $k$ is not a prime.

For binomial $P(x)$ it is equivalent to the existence of infinitely many primes in an arithmetic progression. But I don't know how to go further.



Let $d=\deg P$. Then the $\mathbb Q$-vector space $V=\mathbb Q[X]/(P)$ is $d$-dimensional. The residue classes modulo $(P)$ of $X^2,X^3,X^5,\dots$ are linearly dependent over $\mathbb Q$, so there exist $a_2,a_3,a_5,\ldots\in\mathbb Z$ not all zero such that $a_2X^2+a_3X^3+a_5X^5+\cdots\in(P).$ This shows that there is $Q\in\mathbb Z[X]$, $Q\ne0$ such that $PQ=a_2X^2+a_3X^3+a_5X^5+\cdots$.

  • $\begingroup$ Thank you! It turns out that the "prime" condition is note essential at all. $\endgroup$ – AlgRev Jun 6 '15 at 4:00

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