Special Issues

On sums of four pentagonal numbers with coefficients

  • Received: 01 January 2020
  • Primary: 11B13, 11E25; Secondary: 11D85, 11E20, 11P70

  • 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.

    Citation: Dmitry Krachun, Zhi-Wei Sun. On sums of four pentagonal numbers with coefficients[J]. Electronic Research Archive, 2020, 28(1): 559-566. doi: 10.3934/era.2020029

    Related Papers:

  • 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.



    加载中


    [1] Quaternary quadratic forms representing all integers. Amer. J. Math. (1927) 49: 39-56.
    [2] (1939) Modern Elementary Theory of Numbers.University of Chicago Press.
    [3] Universal sums of generalized pentagonal numbers. Ramanujan J. (2020) 51: 479-494.
    [4] The first nontrivial genus of positive definite ternary forms. Math. Comput. (1995) 64: 341-345.
    [5] Sums of four polygonal numbers with coefficients. Acta Arith. (2017) 180: 229-249.
    [6] A short proof of Cauchy's polygonal theorem. Proc. Amer. Math. Soc. (1987) 99: 22-24.
    [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
    [8] Ramanujan's ternary quadratic form. Invent. Math. (1997) 130: 415-454.
    [9] A result similar to Lagrange's theorem. J. Number Theory (2016) 162: 190-211.
    [10] On universal sums $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$. Nanjing Univ. J. Math. Biquarterly (2018) 35: 85-199.
    [11] Universal sums of three quadratic polynomials. Sci. China Math. (2020) 63: 501-520.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1325) PDF downloads(179) Cited by(1)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog