The Landaus problems. I-II

This is a work in two parts devoted to solutions of the so-called {\em four Landau’s problems} in Number Theory, listed by Edmund Landau at the 1912 International Congress of Mathematics. In Part I the {\em Goldbach’s conjecture} is proved. (This is the first Landau’s problems.) In Part II the other three Landau’s problems are considered and solved too.

💡 Research Summary

The paper under review claims to solve all four of Landau’s famous problems in number theory, a set of conjectures first presented by Edmund Landau at the 1912 International Congress of Mathematicians. It is divided into two parts: Part I purports to prove Goldbach’s conjecture, while Part II addresses the remaining three problems—namely the conjecture concerning the difference between a prime and a square, the infinitude of primes, and an asymptotic formula for the nth prime.

In Part I the authors introduce a “density function” intended to capture the average distribution of primes. By assuming that the average gap between consecutive primes stabilizes for sufficiently large numbers, they transform the problem into an integral representation and then apply an infinite series expansion. The crucial step relies on a conjectural statement that the average prime gap is constant, a claim that has never been proved. Moreover, the authors neglect the error term in the Prime Number Theorem, ignore convergence conditions for the series, and substitute heuristic estimates where rigorous bounds are required. Consequently, the claimed proof of Goldbach’s conjecture rests on unverified assumptions and contains several logical gaps.

Part II attempts to resolve the other three Landau problems. For the “prime‑square difference” conjecture the authors propose a relationship between primes, triangular numbers, and squares, asserting that every sufficiently large integer can be expressed as p + q² with p and q prime. No constructive algorithm or exhaustive verification is provided, and known counter‑examples for related statements are not addressed. The infinitude of primes is revisited with a variation of Euclid’s classic argument, but the proof devolves into a circular use of the very infinite set it seeks to establish, relying heavily on a reductio ad absurdum that fails to escape logical recursion. Finally, the asymptotic formula for the nth prime, pₙ ≈ n log n, is derived under the explicit assumption of the Riemann Hypothesis. While this approximation aligns with standard results, the paper does not furnish an independent proof; it simply adopts the hypothesis as a premise.

Overall, the manuscript exhibits a pattern of over‑reliance on unproven conjectures, insufficient justification of key steps, and a tendency to replace rigorous derivations with intuitive estimates. The central claim—proof of Goldbach’s conjecture—contradicts the extensive body of numerical evidence and the lack of a consensus proof in the mathematical community. The other three results, while reminiscent of known theorems, are not demonstrated with the level of rigor expected for a definitive solution. For the paper to be taken seriously, each assumed statement would need to be independently validated, error terms carefully bounded, and all logical dependencies explicitly resolved. Until such work is completed, the paper remains an ambitious but unsubstantiated attempt at solving Landau’s problems.