The pentagonal theorem of sixty-three and generalizations of Cauchy’s lemma
-
Jangwon Ju
and
Daejun Kim
Abstract
In this paper, we consider the solvability over non-negative integers of certain Diophantine equations coming from representations of integers as sums of pentagonal numbers (counting the number of dots in a regular pentagon). We study a general method to obtain generalized versions of Cauchy’s lemma. Using this, we show the “pentagonal theorem of 63”, which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers
We further show that these integers form a unique minimal universality criterion set.
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: NRF-2022R1A2C1092314
Award Identifier / Grant number: NRF-2020R1A6A3A03037816
Funding source: Korea Institute for Advanced Study
Award Identifier / Grant number: MG085501
Funding statement: The work of the first author was supported by the National Research Foundation of Korea (NRF) grant (No. NRF-2022R1A2C1092314) funded by the Korea government (MSIT). The work of the second author was supported by Basic Science Research Program through NRF funded by the Minister of Education (No. NRF-2020R1A6A3A03037816) and by a KIAS Individual Grant (No. MG085501) at Korea Institute for Advanced Study.
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Articles in the same Issue
- Frontmatter
- Generalized Cauchy–Riemann equations in non-identity bases with application to the algebrizability of vector fields
- Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation
- Minimal Kähler submanifolds in product of space forms
- On the number of rational points of certain algebraic varieties over finite fields
- On two conjectures of Sun concerning Apéry-like series
- A note on Hopf’s lemma and strong minimum principle for nonlocal equations with non-standard growth
- Time-step heat problem on the mesh: asymptotic behavior and decay rates
- Varieties of Borel subalgebras for the Jacobson–Witt Lie algebras
- Ramanujan systems of Rankin–Cohen type and hyperbolic triangles
- Skew-braces and 𝑞-braces
- Products of unipotent elements in certain algebras
- On the Fourier orthonormal bases of a class of self-similar measures on ℝ n
- The pentagonal theorem of sixty-three and generalizations of Cauchy’s lemma
- Restriction estimates in a conical singular space: Schrödinger equation
- Fractional integrals associated with Radon transforms
