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arXiv · 2507.13645

Universal quaternary mixed sums involving generalized 3-, 4-, 5- and 8-gonal numbers via products of Ramanujan's theta functions

Nasser Abdo Saeed Bulkhali·A. Vanitha·M. P. Chaudhary
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Abstract

Generalized $m$-gonal numbers are those $p_m(x)= [ (m - 2)x^2 - (m - 4)x ]/2 $ where $x$ and $m$ are integers with $m \geq 3$. If any nonnegative integer can be written in the form $ap_r(h)+bp_s(l)+cp_t(m)+dp_u(n)$, where $a,b,c,d$ are positive integers, then we call $ap_r(h)+bp_s(l)+cp_t(m)+dp_u(n)$ a universal quaternary sum. In this paper, we determine the universality of many quaternary sums when $r,s,t,u \in \{3,4,5,8\}$, using the theory of Ramanujan's theta function identities

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Nasser Abdo Saeed Bulkhali, A. Vanitha, M. P. Chaudhary. 2025-07-18. Universal quaternary mixed sums involving generalized 3-, 4-, 5- and 8-gonal numbers via products of Ramanujan's theta functions. https://arxiv.org/abs/2507.13645

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