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Existence of best proximity points satisfying two constraint inequalities
On sums of four pentagonal numbers with coefficients
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 |
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.
References:
[1] |
L. E. Dickson,
Quaternary quadratic forms representing all integers, Amer. J. Math., 49 (1927), 39-56.
doi: 10.2307/2370770. |
[2] |
L. E. Dickson, Modern Elementary Theory of Numbers, University of Chicago Press, Chicago, 1939.
![]() |
[3] |
J. Ju,
Universal sums of generalized pentagonal numbers, Ramanujan J., 51 (2020), 479-494.
doi: 10.1007/s11139-019-00142-3. |
[4] |
I. Kaplansky,
The first nontrivial genus of positive definite ternary forms, Math. Comput., 64 (1995), 341-345.
doi: 10.2307/2153338. |
[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. |
[6] |
M. B. Nathanson,
A short proof of Cauchy's polygonal theorem, Proc. Amer. Math. Soc., 99 (1987), 22-24.
doi: 10.2307/2046263. |
[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] |
K. Ono and K. Soundararajan,
Ramanujan's ternary quadratic form, Invent. Math., 130 (1997), 415-454.
doi: 10.1007/s002220050191. |
[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. |
[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. |
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. |
[2] |
L. E. Dickson, Modern Elementary Theory of Numbers, University of Chicago Press, Chicago, 1939.
![]() |
[3] |
J. Ju,
Universal sums of generalized pentagonal numbers, Ramanujan J., 51 (2020), 479-494.
doi: 10.1007/s11139-019-00142-3. |
[4] |
I. Kaplansky,
The first nontrivial genus of positive definite ternary forms, Math. Comput., 64 (1995), 341-345.
doi: 10.2307/2153338. |
[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. |
[6] |
M. B. Nathanson,
A short proof of Cauchy's polygonal theorem, Proc. Amer. Math. Soc., 99 (1987), 22-24.
doi: 10.2307/2046263. |
[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] |
K. Ono and K. Soundararajan,
Ramanujan's ternary quadratic form, Invent. Math., 130 (1997), 415-454.
doi: 10.1007/s002220050191. |
[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. |
[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. |
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