Abstract
Let \(P_y^+(n)\) denote the largest prime factor p of n with \(p\leqslant y\). We prove that there exists a positive proportion of integers n such that \(P_y^+(n)<P_y^+(n+1)\) for \(y=x^{\alpha }\) when \(\alpha \) is small. Especially, the proportion is larger than 1 / 4 when \(\alpha \) tends to 0, which improves our previous result.
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Acknowledgements
Ce travail a été réalisé sous la direction de mes directeurs de thèse Cécile Dartyge et Jie Wu. Je les remercie vivement pour les nombreuses suggestions cruciales qu’ils ont proposées dans l’élaboration de ce travail.
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L’auteur est partiellement soutenu par une bourse de “China Scholarship Council”.
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Wang, Z. Sur les plus grands facteurs premiers inférieur à y d’entiers consécutifs. Ramanujan J 49, 483–490 (2019). https://doi.org/10.1007/s11139-018-0110-z
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DOI: https://doi.org/10.1007/s11139-018-0110-z