### Electronic Research Archive

2020, Issue 1: 559-566. doi: 10.3934/era.2020029
Special Issues

# On sums of four pentagonal numbers with coefficients

• 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.

 [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.
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