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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

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