Landau's problems

Landau's problems are four longstanding unsolved conjectures in number theory related to the distribution and arithmetic properties of prime numbers, which were presented by the German mathematician Edmund Landau during his plenary address at the Fifth International Congress of Mathematicians in Cambridge in 1912.^[1] These problems, often described by Landau himself as "unattackable" with the methods available at the time, have since become foundational challenges in the field, inspiring extensive research in analytic number theory despite remaining unresolved over a century later.^[1] The four conjectures are as follows: Despite significant advances, such as bounds on prime gaps and numerical verifications extending to enormous scales, all four problems remain open as of 2025, underscoring their depth and the limitations of current techniques in prime number theory.^[3]^[1] Landau's presentation not only highlighted these issues but also emphasized their interconnectedness with broader questions about prime distribution, influencing generations of mathematicians.^[4]

Background

Historical presentation

The Fifth International Congress of Mathematicians took place in Cambridge, England, from August 22 to 28, 1912, attracting 574 full members from 27 countries and serving as a major gathering for the global mathematical community.^[5] On August 23, during a plenary session, German mathematician Edmund Landau delivered an address titled "Gelöste und ungelöste Probleme aus der Theorie der Primzahlenverteilung und der Riemannschen Zeta-Funktion" (Solved and Unsolved Problems in the Theory of Prime Number Distribution and the Riemann Zeta Function), in which he outlined four fundamental unsolved problems in analytic number theory related to the distribution of prime numbers.^[6]^[7] The event was attended by prominent figures such as David Hilbert and G.H. Hardy, highlighting its role as a pivotal moment in early 20th-century number theory, just as proofs of the prime number theorem had recently been established.^[5] Edmund Landau (1877–1938), a leading expert in analytic number theory, had already made significant contributions to the field by the time of the congress. In 1903, he provided a simplified proof of the prime number theorem, which describes the asymptotic distribution of primes among the positive integers, building on the independent proofs by Jacques Hadamard and Charles Jean de la Vallée Poussin from 1896.^[6] Additionally, Landau's 1909 handbook, Handbuch der Lehre von der Verteilung der Primzahlen, offered a comprehensive elementary treatment of Dirichlet's theorem on primes in arithmetic progressions, demonstrating his deep engagement with prime distribution problems.^[8] These works positioned Landau as an authoritative voice on the subject, influencing his selection for the plenary address. In his talk, Landau characterized the four problems—Goldbach's conjecture, the twin prime conjecture, Legendre's conjecture, and the conjecture on primes of the form n^2 + 1—as "unattackable" using the methods available at the time, underscoring their foundational importance to understanding prime number distribution despite recent advances like the prime number theorem.^[1] He emphasized that these challenges lay at the core of analytic number theory, resisting the analytic tools that had succeeded elsewhere, and their presentation echoed David Hilbert's famous list of problems from the 1900 congress, inspiring future research directions.^[6] The address, published in the congress proceedings, marked a key historical benchmark, galvanizing efforts in prime number theory for decades.^[7]

Mathematical significance

Landau's problems are intrinsically linked to the prime number theorem, which provides the asymptotic density of primes as \pi(x) \sim \frac{x}{\log x}, by addressing unresolved questions about the finer structure of prime gaps and the frequency of primes in specific sequences such as short intervals or quadratic forms. These conjectures reveal limitations in current knowledge of prime distribution, as even strong error terms in the prime number theorem, such as \pi(x) = \mathrm{Li}(x) + O(x \exp(-c \sqrt{\log x})) for some c > 0, do not suffice to resolve issues like bounded gaps between primes or their occurrence between consecutive squares.^[9] The enduring challenge of these problems has profoundly shaped analytic number theory and sieve methods developed after 1912, including Brun's pure sieve and later refinements like the Selberg sieve, which have been essential for obtaining upper bounds on prime gaps, such as the unconditional bound d_n \ll p_n^{0.525} (Baker–Harman–Pintz, 2001), while under the Riemann hypothesis the bound improves to O(\sqrt{p_n} \log p_n). Efforts to tackle Landau's conjectures have also driven computational techniques, enabling verifications of related statements for enormous ranges, like Goldbach representations up to $4 \times 10^{18}, thereby testing and refining probabilistic models of primes.^[10] Landau's problems extend the tradition of Hilbert's eighth problem, posed in 1900, which sought resolutions to the Riemann hypothesis and the binary Goldbach conjecture as part of broader inquiries into prime distribution. Their influence persists in modern mathematical priorities, paralleling the Clay Mathematics Institute's Millennium Prize Problems—such as the Riemann hypothesis—by motivating interdisciplinary advances in additive number theory, though none are formally included among the Millennium challenges. As of 2025, all four problems remain unsolved, functioning as enduring benchmarks for measuring breakthroughs in prime gap theory and additive bases.^[11]^[3] Beyond pure mathematics, the investigation of Landau's problems has implications for applied fields, enhancing models of prime distributions critical to cryptography for generating secure large primes in systems like RSA, and fostering connections to random matrix theory through analogies between prime spacings and eigenvalue distributions in Gaussian unitary ensembles.^[12]

Goldbach's conjecture

Statement

Goldbach's conjecture, the first of Landau's problems, asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. Formally, for every even integer n > 2, there exist prime numbers p and q such that n = p + q.^[13] The conjecture was first proposed in a 1742 letter from Christian Goldbach to Leonhard Euler, who reformulated it in a stronger version suggesting that every integer greater than 2 is the sum of three primes (now known as the weak Goldbach conjecture). Examples include $4 = 2 + 2, $6 = 3 + 3, $8 = 3 + 5, and $10 = 5 + 5.^[13] This problem highlights the additive properties of primes and connects to broader questions in analytic number theory, though it is independent of the other Landau problems in its specific formulation.^[1]

Progress toward proof

The conjecture remains unproven, but extensive computational verification has confirmed it for all even integers up to $4 \times 10^{18} as of 2013, with subsequent efforts extending checks slightly further but not altering the established limit significantly. No counterexamples have been found, supporting its empirical validity on this scale.^[13]^[14] Significant partial progress includes the weak Goldbach conjecture, which states that every odd integer greater than 5 is the sum of three primes. This was proved by Harald Helfgott in 2013, resolving a problem that had stood since 1742 and providing insights into ternary representations that inform binary cases.^[15] In 1966, Jingrun Chen proved that every sufficiently large even integer is the sum of a prime and a semiprime (a product of at most two primes), known as Chen's theorem. This comes close to the full conjecture by allowing one factor to have two primes but establishes infinitude of such representations.^[16] Using the circle method, Hardy and Littlewood in 1923 conjectured an asymptotic formula for the number of ways r(n) an even n can be written as p + q: r(n) \sim 2 C_2 \frac{n}{(\log n)^2}, where C_2 = \prod_{p>2} \left(1 - \frac{1}{(p-1)^2}\right) \left(1 - \frac{1}{p}\right)^{-2} \approx 0.66016 is the twin prime constant. This heuristic implies the conjecture holds for almost all even numbers and has been numerically supported.^[13] Further results show that every even integer greater than 2 is the sum of at most four primes (Schnirelmann, 1930s, via Vinogradov's theorem) and that the number of primes needed is bounded by six unconditionally. Sieve methods and the Hardy-Littlewood approach have yielded error terms and density estimates, but barriers in handling the singularity at small primes prevent a full proof as of 2025.^[13] The problem's difficulty stems from the irregularity of prime distribution, with ongoing research focusing on refined sieve techniques and connections to the Riemann hypothesis, though no resolution is in sight.

Twin prime conjecture

Statement

The twin prime conjecture, the second of Landau's problems, asserts that there are infinitely many pairs of prime numbers that differ by 2, called twin primes. Formally, there are infinitely many primes p such that p + 2 is also prime.^[1] Examples include (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), and (59, 61).^[17] The conjecture was first proposed in a more general form by Alphonse de Polignac in 1849 and later emphasized by Edmund Landau in his 1912 address at the International Congress of Mathematicians.^[2]

Progress toward proof

Viggo Brun developed sieve methods in the early 20th century, showing that the sum of reciprocals over twin prime pairs converges to Brun's constant, approximately 1.902160583, indicating that twin primes have zero density among all primes.^[17] In 1973, Jing-Run Chen proved that there are infinitely many primes p such that p + 2 is either prime or the product of two primes (a semiprime), establishing the infinitude of "almost twin primes."^[1] A major breakthrough occurred in 2013 when Yitang Zhang proved that there are infinitely many pairs of primes differing by at most 70 million, using a novel sieve theory approach combined with Bombieri-Vinogradov-type estimates.^[18] Subsequent improvements by the Polymath8 project reduced this bound to 246 by 2014, confirming infinitely many prime pairs with gaps at most 246. Independently, James Maynard achieved a bound of 600 using a multidimensional sieve method focused on admissible sets of linear forms. These results do not resolve the twin prime case (gap exactly 2) but demonstrate bounded gaps for some even differences.^[19]^[20] Under stronger assumptions like the Elliott-Halberstam conjecture, the bound can be reduced to 12 or even 6, but the full twin prime conjecture remains open as of 2025.^[21] Computational verification has confirmed the conjecture for all even numbers up to beyond $10^{18}, with the largest known twin prime pair as of January 2025 being $2996863034895 \times 2^{1290000} \pm 1, which has 388,342 digits. Ongoing searches continue to extend these records using distributed computing projects.^[22] Challenges persist due to the sieve level limitations and the need to isolate the exact gap of 2, with connections to the distribution of primes in arithmetic progressions and zero-density estimates in the critical strip.

Legendre's conjecture

Statement

Legendre's conjecture, the third of Landau's four problems, states that for every positive integer n, there is at least one prime number p such that n^2 < p < (n+1)^2.^[23] This conjecture was first proposed by Adrien-Marie Legendre in 1808 in his work on the distribution of primes.^[23] It was later highlighted by Edmund Landau in his 1912 address at the International Congress of Mathematicians.^[1] Examples of such primes include: for n=1, p=2 (between 1 and 4); for n=2, p=5 (between 4 and 9); for n=3, p=7 or $11 (between 9 and 16); and for n=4, p=17 (between 16 and 25).^[23] The conjecture is a special case of more general questions about prime gaps and the distribution of primes in short intervals. It implies that the gap between consecutive squares contains at least one prime, with the interval length being $2n+1.

Progress toward proof

Legendre's conjecture remains unproven as of 2025, but significant partial results and computational verifications support its plausibility. In 1975, Jingrun Chen proved that for sufficiently large n, there is always at least one prime or semiprime (product of two primes) in the interval (n^2, (n+1)^2). This result uses sieve methods and provides a weaker but unconditional affirmation of the existence of numbers with few prime factors in these intervals.^[23] The prime number theorem implies that the number of primes up to x is approximately x / \log x, suggesting about $2n / \log(n^2) \approx 1 / \log n primes in the interval, which is greater than 1 for large n, but this is asymptotic and does not guarantee at least one prime for every n. Under the Riemann hypothesis, the conjecture follows from stronger bounds on the error term in the prime number theorem, ensuring a prime in intervals of length about n^{1/2 + \epsilon}. However, this conditional result does not resolve the unconditional case. Computational verifications have confirmed the conjecture for all n up to approximately $7 \times 10^{13} as of 2024, using efficient algorithms to check prime existence in subintervals.^[24] Further progress includes results on prime gaps, such as those by Iwaniec and Pintz (1984), who showed that there is always a prime between n - n^{0.525} and n for large n, which overlaps with Legendre intervals but does not fully prove it. Sieve techniques and elliptic curve methods continue to extend computational bounds, but theoretical proof remains elusive due to challenges in controlling the distribution of primes in quadratic-length intervals.^[23]

Primes of the form n² + 1

Statement

The fourth of Landau's problems concerns the infinitude of primes of the form n^2 + 1. Formally, the conjecture states that there are infinitely many positive integers n such that n^2 + 1 is a prime number.^[1] Examples of such primes include $2forn=1

, &#36;5

for n=2, $17forn=4

, &#36;37

for n=6, $101forn=10

, and &#36;197

for n=14.^[1] This question was considered by Leonhard Euler in the 18th century and later highlighted by Edmund Landau as the fourth problem in his address at the 1912 International Congress of Mathematicians.^[21]^[9] The conjecture forms a special case of Bunyakovsky's conjecture from 1857, which asserts that a polynomial f(n) \in \mathbb{Z} of degree greater than $1, irreducible over the integers, with positive leading coefficient, and such that the values f(1), f(2), \dotshave no fixed prime divisor, produces infinitely many primes asnvaries over the positive integers. Forf(n) = n^2 + 1$, the polynomial is irreducible over the integers (as a primitive quadratic with no rational roots) and satisfies the other conditions.^[25]

Progress toward proof

In 1923, Hardy and Littlewood conjectured that the number of primes of the form n^2 + 1 with n \leq X is asymptotically C \int_2^X \frac{dx}{\log x}, where C = \prod_{p \equiv 3 \pmod{4}} \left(1 - \frac{1}{p-1}\right) is a positive constant approximately equal to 0.686. This conjecture extends their circle method framework to binary quadratic forms and implies infinitude, though it remains unproven. Significant progress came in 1978 when Iwaniec applied sieve methods to show that there are infinitely many positive integers n such that n^2 + 1 is either prime or a semiprime (the product of two primes). This result establishes the infinitude of values of n^2 + 1 with at most two prime factors but falls short of proving infinitude for primes alone, as the sieve captures "almost-primes" rather than exact primes. A related breakthrough by Friedlander and Iwaniec in 1998 demonstrated that there are infinitely many primes of the form n^2 + m^4, using an asymptotic sieve tailored to this binary form, which shares structural similarities with n^2 + 1 but allows for a higher-dimensional sifting level. This theorem highlights advances in representing primes by irreducible polynomials, though it does not directly resolve the case for m = 1. Sieve techniques, bolstered by the Bombieri-Vinogradov theorem, have shown that n^2 + 1 has few prime factors on average for n up to large X, with the theorem providing level-of-distribution estimates that control error terms in the distribution of primes in arithmetic progressions relevant to quadratic residues. These methods bound the exceptional set where n^2 + 1 is highly composite, supporting the heuristic density but insufficient for isolating primes due to limitations in the sifting dimension for this quadratic. In 2019, it was shown that the largest prime factor of n^2 + 1 exceeds n^{1.279} for infinitely many n.^[26] Computational efforts have identified primes of the form n^2 + 1 for n exceeding $10^6, with ongoing searches extending records, yet no theoretical proof of infinitude exists as of 2025.^[27] The problem faces challenges from local conditions, such as n^2 + 1 always being odd and thus avoiding divisibility by 2, but global density issues persist, including connections to class number problems in quadratic fields like \mathbb{Q}(i), where the representation as norms complicates asymptotic counts. Unlike the twin prime conjecture, no bounded gap analogs have been established, and sieve barriers prevent reaching the prime level. As of 2025, this remains the least progressed of Landau's problems, with Heath-Brown's work on primes represented by binary cubic forms, such as x^3 + 2y^3, providing indirect insights into polynomial prime representations through circle method refinements applicable to higher-degree analogs.

Landau's problems

Background

Historical presentation

Mathematical significance

Goldbach's conjecture

Statement

Progress toward proof

Twin prime conjecture

Statement

Progress toward proof

Legendre's conjecture

Statement

Progress toward proof

Primes of the form n² + 1

Statement

Progress toward proof

References