Claims

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2001_05_01_heath_brown: Heath-Brown proves that a positive proportion of n in (X, 2X] have n^3+2 with a prime factor above X^(1+10^-303), so the answer is yes for t^3+2.

2014_11_28_irving: Irving proves that a positive proportion of n in (X, 2X] have n^3+2 with a prime factor above X^(1+10^-52), improving Heath-Brown's exponent for t^3+2.

2015_06_22_de_la_breteche: De la Bretèche proves that for every even monic irreducible quartic with Galois group V4 a positive density of n <= X have f(n) with a prime factor above X^(1+c), so the answer is yes for these quartics.

2022_12_07_dartyge_maynard: For a monic irreducible quartic with Galois group C4 or D4, a positive proportion of m in (x, 2x] have f(m) with a prime factor at least x^(1+c), so the answer is yes for these quartics.

2023_08_19_carella: An arXiv note claims that the greatest prime factor of the product of n^2+1 over x <= n <= 2x is at least x^(3/2) as x tends to infinity.

2024_04_05_pascadi: The proof of Pascadi's Theorem 1.1 bounds the greatest prime factor of the product of m^2+1 over x <= m <= 2x below by x^1.30008 for all large x, which answers the first question for t^2+1.

2025_05_01_grimmelt_merikoski: An arXiv preprint gives, for fixed coprime a and squarefree h, an m in [X, 2X] with a prime factor of am^2+h above X^1.312 for all large X, so the answer would be yes for these quadratics.

2026_02_03_ermoshin: An arXiv preprint proves that for every monic irreducible cubic a positive proportion of m in [x, 2x] have f(m) with a prime factor above x^(1+c), so the answer would be yes for these cubics.

2026_04_16_bhalla: An unpublished note derives the degree-scale bound, the greatest prime factor of the running product at least a constant times n to the degree, for every irreducible polynomial from an unproved prime-values hypothesis.

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