American Institute of Mathematical Sciences

Advanced
March  2020, 28(1): 559-566. doi: 10.3934/era.2020029

On sums of four pentagonal numbers with coefficients

Dmitry Krachun 1, and  Zhi-Wei Sun 2,,

1. 

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, 191023, St. Petersburg, Russia
2. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Zhi-Wei Sun

Received  January 2020 Published  March 2020

Fund Project: The work is supported by the NSFC(Natural Science Foundation of China)-RFBR(Russian Foundation for Basic Research) Cooperation and Exchange Program (grants NSFC 11811530072 and RFBR 18-51-53020-GFEN-a). The second author is also supported by the Natural Science Foundation of China (grant no. 11971222)

The pentagonal numbers are the integers given by$ p_5(n) = n(3n-1)/2\ (n = 0,1,2,\ldots) $.Let $ (b,c,d) $ be one of the triples $ (1,1,2),(1,2,3),(1,2,6) $ and $ (2,3,4) $.We show that each $ n = 0,1,2,\ldots $ can be written as $ w+bx+cy+dz $ with $ w,x,y,z $ pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integeris a sum of five pentagonal numbers two of which are equal; this refines a classical resultof Cauchy claimed by Fermat.

Keywords: Pentagonal numbers, additive bases, ternary quadratic forms.
Mathematics Subject Classification: Primary: 11B13, 11E25; Secondary: 11D85, 11E20, 11P70.
Citation: Dmitry Krachun, Zhi-Wei Sun. On sums of four pentagonal numbers with coefficients. Electronic Research Archive, 2020, 28 (1) : 559-566. doi: 10.3934/era.2020029
References:
[1]

L. E. Dickson, Quaternary quadratic forms representing all integers, Amer. J. Math., 49 (1927), 39-56.  doi: 10.2307/2370770.  Google Scholar

[2] L. E. Dickson, Modern Elementary Theory of Numbers, University of Chicago Press, Chicago, 1939.   Google Scholar
[3]

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

[4]

I. Kaplansky, The first nontrivial genus of positive definite ternary forms, Math. Comput., 64 (1995), 341-345.  doi: 10.2307/2153338.  Google Scholar

[5]

X.-Z. Meng and Z.-W. Sun, Sums of four polygonal numbers with coefficients, Acta Arith., 180 (2017), 229-249.  doi: 10.4064/aa8630-4-2017.  Google Scholar

[6]

M. B. Nathanson, A short proof of Cauchy's polygonal theorem, Proc. Amer. Math. Soc., 99 (1987), 22-24.  doi: 10.2307/2046263.  Google Scholar

[7]

M. B. Nathanson, Additive Number Theory: The Classical Bases, Grad. Texts in Math., vol. 164, Springer, New York, 1996. doi: 10.1007/978-1-4757-3845-2.  Google Scholar

[8]

K. Ono and K. Soundararajan, Ramanujan's ternary quadratic form, Invent. Math., 130 (1997), 415-454.  doi: 10.1007/s002220050191.  Google Scholar

[9]

Z.-W. Sun, A result similar to Lagrange's theorem, J. Number Theory, 162 (2016), 190-211.  doi: 10.1016/j.jnt.2015.10.014.  Google Scholar

[10]

Z.-W. Sun, On universal sums $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$, Nanjing Univ. J. Math. Biquarterly, 35 (2018), 85-199.   Google Scholar

[11]

Z.-W. Sun, Universal sums of three quadratic polynomials, Sci. China Math., 63 (2020), 501-520.  doi: 10.1007/s11425-017-9354-4.  Google Scholar

show all references

References:
[1]

L. E. Dickson, Quaternary quadratic forms representing all integers, Amer. J. Math., 49 (1927), 39-56.  doi: 10.2307/2370770.  Google Scholar

[2] L. E. Dickson, Modern Elementary Theory of Numbers, University of Chicago Press, Chicago, 1939.   Google Scholar
[3]

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

[4]

I. Kaplansky, The first nontrivial genus of positive definite ternary forms, Math. Comput., 64 (1995), 341-345.  doi: 10.2307/2153338.  Google Scholar

[5]

X.-Z. Meng and Z.-W. Sun, Sums of four polygonal numbers with coefficients, Acta Arith., 180 (2017), 229-249.  doi: 10.4064/aa8630-4-2017.  Google Scholar

[6]

M. B. Nathanson, A short proof of Cauchy's polygonal theorem, Proc. Amer. Math. Soc., 99 (1987), 22-24.  doi: 10.2307/2046263.  Google Scholar

[7]

M. B. Nathanson, Additive Number Theory: The Classical Bases, Grad. Texts in Math., vol. 164, Springer, New York, 1996. doi: 10.1007/978-1-4757-3845-2.  Google Scholar

[8]

K. Ono and K. Soundararajan, Ramanujan's ternary quadratic form, Invent. Math., 130 (1997), 415-454.  doi: 10.1007/s002220050191.  Google Scholar

[9]

Z.-W. Sun, A result similar to Lagrange's theorem, J. Number Theory, 162 (2016), 190-211.  doi: 10.1016/j.jnt.2015.10.014.  Google Scholar

[10]

Z.-W. Sun, On universal sums $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$, Nanjing Univ. J. Math. Biquarterly, 35 (2018), 85-199.   Google Scholar

[11]

Z.-W. Sun, Universal sums of three quadratic polynomials, Sci. China Math., 63 (2020), 501-520.  doi: 10.1007/s11425-017-9354-4.  Google Scholar

[1]

Anish Ghosh, Dubi Kelmer. A quantitative Oppenheim theorem for generic ternary quadratic forms. Journal of Modern Dynamics, 2018, 12: 1-8. doi: 10.3934/jmd.2018001
[2]

Dubi Kelmer. Quadratic irrationals and linking numbers of modular knots. Journal of Modern Dynamics, 2012, 6 (4) : 539-561. doi: 10.3934/jmd.2012.6.539
[3]

Shrikrishna G. Dani. Simultaneous diophantine approximation with quadratic and linear forms. Journal of Modern Dynamics, 2008, 2 (1) : 129-138. doi: 10.3934/jmd.2008.2.129
[4]

Constantin N. Beli. Representations of integral quadratic forms over dyadic local fields. Electronic Research Announcements, 2006, 12: 100-112.

[5]

Jyrki Lahtonen, Gary McGuire, Harold N. Ward. Gold and Kasami-Welch functions, quadratic forms, and bent functions. Advances in Mathematics of Communications, 2007, 1 (2) : 243-250. doi: 10.3934/amc.2007.1.243
[6]

Shigeki Akiyama, Edmund Harriss. Pentagonal domain exchange. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4375-4400. doi: 10.3934/dcds.2013.33.4375
[7]

Víctor Guíñez, Eduardo Sáez. Versal Unfoldings for rank--2 singularities of positive quadratic differential forms: The remaining case. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 887-904. doi: 10.3934/dcds.2005.12.887
[8]

Jiyoung Han, Seonhee Lim, Keivan Mallahi-Karai. Asymptotic distribution of values of isotropic here quadratic forms at S-integral points. Journal of Modern Dynamics, 2017, 11: 501-550. doi: 10.3934/jmd.2017020
[9]

Ye Tian, Shu-Cherng Fang, Zhibin Deng, Wenxun Xing. Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming. Journal of Industrial & Management Optimization, 2013, 9 (3) : 703-721. doi: 10.3934/jimo.2013.9.703
[10]

Murray R. Bremner. Polynomial identities for ternary intermolecular recombination. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1387-1399. doi: 10.3934/dcdss.2011.4.1387
[11]

Ismara  Álvarez-Barrientos, Mijail Borges-Quintana, Miguel Angel Borges-Trenard, Daniel Panario. Computing Gröbner bases associated with lattices. Advances in Mathematics of Communications, 2016, 10 (4) : 851-860. doi: 10.3934/amc.2016045
[12]

Peng Sun. Exponential decay of Lebesgue numbers. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773
[13]

Danny Calegari, Alden Walker. Ziggurats and rotation numbers. Journal of Modern Dynamics, 2011, 5 (4) : 711-746. doi: 10.3934/jmd.2011.5.711
[14]

Xavier Buff, Nataliya Goncharuk. Complex rotation numbers. Journal of Modern Dynamics, 2015, 9: 169-190. doi: 10.3934/jmd.2015.9.169
[15]

Tatsuya Maruta, Yusuke Oya. On optimal ternary linear codes of dimension 6. Advances in Mathematics of Communications, 2011, 5 (3) : 505-520. doi: 10.3934/amc.2011.5.505
[16]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
[17]

Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012
[18]

Piotr Pokora, Tomasz Szemberg. Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone. Electronic Research Announcements, 2014, 21: 126-131. doi: 10.3934/era.2014.21.126
[19]

Seok-Jin Kang and Jae-Hoon Kwon. Quantum affine algebras, combinatorics of Young walls, and global bases. Electronic Research Announcements, 2002, 8: 35-46.

[20]

Anna Chiara Lai, Paola Loreti. Robot's finger and expansions in non-integer bases. Networks & Heterogeneous Media, 2012, 7 (1) : 71-111. doi: 10.3934/nhm.2012.7.71

2018 Impact Factor: 0.263

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences