en.wikipedia.org/wiki/Hilbert%27s_tenth_problem
2 corrections found
The problem for the ring of integers of algebraic number fields other than those covered by the results above remains open.
This is outdated. A 2026 peer-reviewed result proves Hilbert’s tenth problem has a negative answer for the ring of integers of any number field, so the remaining algebraic-number-field cases are no longer open.
Full reasoning
This sentence is no longer correct as of March 2026. In the peer-reviewed paper Rank stability in quadratic extensions and Hilbert’s tenth problem for the ring of integers of a number field (Inventiones Mathematicae, 2026), Alpöge, Bhargava, Ho, and Shnidman state:
- “Hilbert's tenth problem over the ring of integers of any number field has a negative solution.”
- Equivalently, for the ring (\mathcal O_K) of integers of any number field (K), “there does not exist an algorithm” deciding solvability of polynomial equations over (\mathcal O_K).
That directly contradicts the claim that the cases beyond the previously known families “remain open.” The article’s next sentence about the rationals remaining open is still compatible with this newer result, because it concerns (\mathbb Q), not rings of integers of number fields.
2 sources
- Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field | Inventiones mathematicae | Springer Nature Link
Abstract: ... This result implies that Hilbert's tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring O_K of integers of any number field K, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over O_K has solutions in O_K.
- [2501.18774] Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field
Abstract: ... This result implies that Hilbert's tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring O_K of integers of any number field K, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over O_K has solutions in O_K.
(On the other hand, no polynomial can only take on prime values.)
This is too broad as written. The standard impossibility result is for nonconstant polynomials; a constant polynomial such as q(x)=2 takes only the prime value 2.
Full reasoning
The sentence is literally false because constant polynomials are polynomials. For example, the constant polynomial (q(x)=2) takes only the prime value 2.
What is true is the familiar stronger-looking statement with an extra qualifier: no nonconstant polynomial with integer coefficients can output a prime for every integer input. A standard proof of that fact is given in sources on prime-generating formulas, but the article’s wording omits the essential word nonconstant.
2 sources
- Formula for primes - Wikipedia
It is known that no non-constant polynomial function P(n) with integer coefficients exists that evaluates to a prime number for all integers n.
- constant polynomial in nLab
A constant polynomial is a polynomial ... A constant polynomial of R[S] is a polynomial which is in the image of the canonical ring homomorphism h : R → R[S].