Let n be a positive integer greater than 1 and $p_n(x)$ be a polynomial of degree $n$ such that $p(0)= 2, p(1)=3, p(2)=5$, and in general $p(k)$ is the $(k+1)$th prime, for $0\leq k \leq n$.
Is $p_n(x)$ irreducible in $\mathbb Q [x]$?
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2$\begingroup$ Since the $p_n$ may have non-integral coefficients, the factors do not have to lie in $\mathbb{Z}[X]$. $\endgroup$– R.P.Jan 28, 2014 at 12:39
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3$\begingroup$ I checked it for $n\le100$: they are all irreducible! $\endgroup$– Alex DegtyarevJan 28, 2014 at 12:53
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1 Answer
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"Most" polynomials are irreducible. More precisely, in this case, the heights of the coefficients of $p_n(x)$ are (at least) exponential in $n$. Among all polynomials of degree $n$ whose coefficients have height on the order of $C^n$ for some fixed $C>1$, only a tiny proportion will be reducible. So once you've checked that $p_n(x)$ is irreducible for the first few $n$, say $n\le10$, it then becomes likely that the rest of them will be irreducible. But I doubt that this has anything to do with the primes. If you took some other "random looking" sequence of integers $a_k$ that grows appropriately and set $p_n(k)=a_k$ for $0\le k\le n$, you should get similar properties. For example, something weird like $a_k=\lfloor \pi (k+1) \log(k+2)\rfloor$.
answered Jan 28, 2014 at 14:06
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5$\begingroup$ This does not seem to answer the question about this particular sequence. Some polynomials are still reducible. $\endgroup$ Jan 28, 2014 at 14:18
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8$\begingroup$ @AlexDegtyarev Yes, of course, lots of polynomials are reducible. My point is that if you pick a sequence of polynomials with large coefficients more-or-less at random, then it's not surprising that they're all irreducible. So I don't see why the posed question is interesting unless the OP has some particular application in mind that would follow from knowing that his $p_n$'s are irreducible. And if he does have such an application in mind, then it would be good to include it as part of his post. $\endgroup$ Jan 28, 2014 at 14:43
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$\begingroup$ Reducible polynomials tend to take composite values, so prime values may be important. $\endgroup$ Apr 6, 2016 at 11:06