2022/12/07 by Sai Teja Somu, Somu, Sai Teja, Ting Hon Stanford Li +3
Mathematics · #11B83 (Primary) #11D85 (Secondary) #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2212.03673
openalex publication_date 2022/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A positive integer n is said to be a practical number if every integer in [1,n] can be represented as the sum of distinct divisors of n. In this article, we consider practical numbers of a given polynomial form. We give a necessary and sufficient condition on coefficients a and b for there to be infinitely many practical numbers of the form an+b. We also give a necessary and sufficient for a quadratic polynomial to contain infinitely many practical numbers, using which we solve first part of a conjecture mentioned in [9]. In the final section, we prove that every number of 8k+1 form can be expressed as a sum of a practical number and a square, and for every j∈ \0,…,7\∖ \1\ there are infinitely many natural numbers of 8k+j form which cannot be written as sum of a square and a practical number.