In a recent breakthrough, Pascadi (Pascadi 2026) proved that the greatest prime factor of exceeds for infinitely many integers . This is the strongest result to date toward the famous fourth problem of Landau, which predicts that the polynomial takes prime values infinitely often.
Many of the deepest achievements in modern analytic number theory can be viewed as instances of the following overarching problem.
This formulation encompasses a remarkable range of classical and modern results.
The story begins in 1837 with Dirichlet’s theorem on arithmetic progressions: if and are coprime integers, then the linear polynomial takes prime values for infinitely many integers . This completely resolves Question in the case of a single linear polynomial in one variable. It already illustrates a key principle: a simple necessary condition (here, ) is also sufficient.
The next layer of complexity appears with quadratic polynomials. Fermat asserted—and Euler proved—that every prime can be written as for suitable integers . Combined with Dirichlet’s theorem, this shows that the quadratic form represents infinitely many primes.
Euler also established that there are infinitely many primes of the form and . Later, Dirichlet settled the case of Question for binary quadratic forms and Iwaniec extended this to all irreducible quadratic polynomials in two variables that genuinely depend on both variables.
For polynomials of degree at least , progress proved much more difficult. A landmark result came in 1998, when Friedlander and Iwaniec (Friedlander and Iwaniec 1998) showed that the quartic form takes prime values infinitely often. This was the first example of a genuinely nonlinear, higher-degree polynomial in two variables known to produce infinitely many primes.
Shortly thereafter, Heath-Brown (Heath-Brown 2001) established the same for and Heath-Brown and Moroz (Heath-Brown and Moroz 2002) extended the method to general irreducible cubic forms with , provided that the form is not identically even. These works introduced powerful refinements of the Hardy–Littlewood circle method and sieve techniques.
In 2017, Heath-Brown and Li (Heath-Brown and Li 2017) proved that there are infinitely many primes of the form where is an integer and is prime. This answers Question for the pair of polynomials requiring both and to be prime.
A dramatic advance occurred in 2025, when Green and Sawhney (Green and Sawhney 2024) proved the first result in which both variables are required to be prime.
They further observed that the same method should apply to general positive definite binary quadratic forms with , provided that there is no fixed prime divisor of . This effectively settles Question for three polynomials under the stated hypotheses.
Question also encompasses longer patterns of primes. In 2008, Green and Tao (Green and Tao 2008) proved that for every there are infinitely many arithmetic progressions of length consisting entirely of primes. This settles Question for with .
In the same year, Tao and Ziegler (Tao and Ziegler 2008) extended this to polynomial progressions for arbitrary integer polynomials satisfying . These results required a profound synthesis of additive combinatorics, ergodic theory, and analytic number theory.
Despite this extraordinary progress, many elementary-looking instances of Question remain unresolved.
The case , is the twin prime conjecture.
The case , is the conjecture on Sophie Germain primes.
Even the single-polynomial case , is open for every nonlinear polynomial .
A particularly famous example is Landau’s fourth problem, which asks whether is prime for infinitely many integers .
Although we cannot yet prove that is prime infinitely often, we can study its factorization. In 2010, Friedlander and Iwaniec (Friedlander and Iwaniec 2010, Theorem 25.9) showed that is the product of two primes for infinitely many . Obviously, the larger of two prime factors must be exceed . In fact, much earlier, Chebyshev (with the first published proof due to Markov in 1895) proved that for every constant , the greatest prime factor of exceeds for infinitely many . Over more than a century, this linear bound was steadily improved.
Before 2026, the strongest result was due to Merikoski (Merikoski 2023), who proved that the greatest prime factor exceeds infinitely often. Pascadi (Pascadi 2026) has now pushed this further.
While this still falls short of proving that is itself prime infinitely often, it represents the most substantial quantitative progress so far toward Landau’s conjecture.