Universal sums of generalized pentagonal numbers

Abstract

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.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Bhargava, M.: On the Conway–Schneeberger fifteen theorem. Contemp. Math. 272, 27–38 (2000)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bhargava, M., Hanke, J.: Universal quadratic forms and the 290 theorem. Invent. Math. (to appear)

  3. 3.

    Bosma, W., Kane, B.: The triangular theorem of eight and representation by quadratic polynomials. Proc. Am. Math. Soc. 141, 1473–1486 (2013)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Dickson, L.E.: Quaternary quadratic forms representing all integers. Am. J. Math. 49, 39–56 (1927)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Jagy, W.C.: Five regular or nearly-regular ternary quadratic forms. Acta Arith. 77, 361–367 (1996)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Jones, B.W.: Representation by Positive Ternary Quadratic Forms. Unpublished PhD dissertation, University of Chicago (1928)

  7. 7.

    Ju, J., Oh, B.-K.: Universal sums of generalized octagonal numbers. J. Number Theory 190, 292–302 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Ju, J., Oh, B.-K., Seo, B.: Ternary universal sums of generalized polygonal numbers. Int. J. Number Theory (to appear)

  9. 9.

    Kitaoka, Y.: Arithmetic of Quadratic Forms. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  10. 10.

    Oh, B.-K.: Regular positive ternary quadratic forms. Acta Arith. 147, 233–243 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Oh, B.-K., Yu, H.: Completely \(p\)-primitive binary quadratic forms. J. Number Theory 193, 373–385 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Oh, B.-K.: Ternary universal sums of generalized pentagonal numbers. J. Korean Math. Soc. 48, 837–847 (2011)

    MathSciNet  Article  Google Scholar 

  13. 13.

    O’Meara, O.T.: Introduction to Quadratic Forms. Springer, New York (1963)

    Google Scholar 

  14. 14.

    Sun, Z.-W.: A result similar to Lagrange’s theorem. J. Number Theory 162, 190–211 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Sun, Z.-W.: On universal sums of polygonal numbers. Sci. China Math. 58, 1367–1396 (2015)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jangwon Ju.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Generalized pentagonal numbers
  • Pentagonal theorem of 109

Mathematics Subject Classification

  • 11E12
  • 11E20