Universal sums of generalized pentagonal numbers


For an integer x, an integer of the form \(P_5(x)=\frac{3x^2-x}{2}\) is called a generalized pentagonal number. For positive integers \(\alpha _1,\dots ,\alpha _k\), a sum \(\Phi _{\alpha _1,\dots ,\alpha _k}(x_1,x_2,\dots ,x_k)=\alpha _1P_5(x_1)+\alpha _2P_5(x_2)+\cdots +\alpha _kP_5(x_k)\) of generalized pentagonal numbers is called universal if \(\Phi _{\alpha _1,\dots ,\alpha _k}(x_1,x_2,\dots ,x_k)=N\) has an integer solution \((x_1,x_2,\dots ,x_k) \in {\mathbb {Z}}^k\) for any non-negative integer N. In this article, we prove that there are exactly 234 proper universal sums of generalized pentagonal numbers. Furthermore, the “pentagonal theorem of 109” is proven, which states that an arbitrary sum \(\Phi _{\alpha _1,\dots ,\alpha _k}(x_1,x_2,\dots ,x_k)\) is universal if and only if it represents the integers 1, 3, 8, 9, 11, 18, 19, 25, 27, 43, 98, and 109.

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Correspondence to Jangwon Ju.

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This work was supported by the 2019 Research Fund of University of Ulsan.

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Ju, J. Universal sums of generalized pentagonal numbers. Ramanujan J 51, 479–494 (2020). https://doi.org/10.1007/s11139-019-00142-3

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  • Generalized pentagonal numbers
  • Pentagonal theorem of 109

Mathematics Subject Classification

  • 11E12
  • 11E20