Extending Computational Verification of Lemoine’s Conjecture to 1500 Digit Odd Numbers

Duncan Ndegwa; Loyford Njagi; Stephen Luketero; Benard Nzimbi; Kikwai Benjamin

doi:https://doi.org/10.14445/22315373/IJMTT-V72I3P106

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 3 | Year 2026 | Article Id. IJMTT-V72I3P106 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I3P106

Extending Computational Verification of Lemoine’s Conjecture to 1500 Digit Odd Numbers

Duncan Ndegwa, Loyford Njagi, Stephen Luketero, Benard Nzimbi, Kikwai Benjamin

Received	Revised	Accepted	Published
19 Jan 2026	24 Feb 2026	15 Mar 2026	28 Mar 2026

Citation :

Duncan Ndegwa, Loyford Njagi, Stephen Luketero, Benard Nzimbi, Kikwai Benjamin, "Extending Computational Verification of Lemoine’s Conjecture to 1500 Digit Odd Numbers," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 3, pp. 49-52, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I3P106

Abstract

Lemoine’s conjecture claims that every odd integer greater than 5 can be represented as the sum of an odd prime and twice another prime. Recent computational studies have verified the conjecture up to large numerical bounds using deterministic and probabilistic primality testing methods. In this work, we introduce a novel mathematical decomposition algorithm that enables scalable verification of the conjecture for random odd integers of up to 1500 digits. This algorithm constructs candidate partitions 𝑂=𝑝+2𝑞 by iterating over a subset of odd primes 𝑝<𝑂, testing whether 𝑞=(𝑂−𝑝)/2 is prime. This approach narrows the search space and is tailored for extremely large integers. The primality of candidates is tested using the Miller-Rabin probabilistic method, supporting efficient computation. We demonstrate the algorithm’s effectiveness across a range of large-digit odd numbers—the implementation, though secondary, was done in Python to validate the theoretical method. Our results extend the conjecture’s verification range and establish a foundation for future theoretical and computational exploration.

Keywords

Lemoine’s Conjecture, Prime Decomposition, Odd Integer Representation, Primality Testing.

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