Analytical Approach Toward the Distribution of Near Square Primes in Relation to Landau's Fourth Problem

Abhijit Manohar

Abstract: The 4th Landau's problem also known as the 'near square primes' asked whether there are infinitely many primes which are of the form p = n² + 1, n ∈ N. The solution is yet unresolved and this article attempts to present the solution. This article explores the fourth of Landau's problems, which asks whether there are infinitely many prime numbers of the form n² + 1. Using tabular data and graphical analysis, the paper investigates the distribution patterns and last-digit trends of such primes, referred to as near square primes. It presents numerical evidence suggesting that although the density of these primes declines with increasing n, they do not cease to occur, thereby implying an infinite continuation. The study employs quadratic modeling and logarithmic regressions to support this hypothesis.

Keywords: near snear square primes, Landau's problems, prime number distribution, quadratic equations, number theoryquare primes, Landau's problems, prime number distribution, quadratic equations, number theory

How to Cite?: Abhijit Manohar, "Analytical Approach Toward the Distribution of Near Square Primes in Relation to Landau's Fourth Problem", Volume 14 Issue 12, December 2025, International Journal of Science and Research (IJSR), Pages: 790-794, https://www.ijsr.net/getabstract.php?paperid=SR251210104407, DOI: https://dx.doi.org/10.21275/SR251210104407