Skip to content
DraconiticResearch
Search Sources API
186,541 studies31 sourcesUpdated 56 min ago
PreprintarXiv licencearXiv

Computational results on sums of a prime with squares or cubes

  • Kenny Applegate
  • Kyle Pratt
Open at sourcePDFDOICite

Abstract

One conjectures that all integers greater than 14 are the sum of an odd prime and two positive squares. We prove that all integers n with 14<n<1026 can be represented in this manner. Along the way, we investigate sums of a prime and one square and then ``bootstrap'' to get our results on sums of a prime and two squares. We also show that all integers between $308$ and 1016 can be represented as the sum of an odd prime and two positive cubes. Our methods require the determination of the integral points on various elliptic curves.

Related

  1. Space-filling surfaces: sharp Hölder continuous parameterizations from squares to cubes
    Matthew Badger, Kevin Palmer
    arXivPreprintarXiv licenceFull textAnalysis & PDEsGeometry & topologyarXiv
  2. The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes
    Manjul Bhargava
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  3. The large sieve inequality and additive decompositions of sums of squares
    Christian Elsholtz, Julia Stadlmann
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Density Bounds in Hua's Theorem for Prime Squares
    Michael T. Lacey, Hamed Mousavi, Yaghoub Rahimi +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. On the Distribution of Prime Squares and Its Applications
    Hamed Mousavi
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{applegate2026computational,
  title = {{Computational results on sums of a prime with squares or cubes}},
  author = {Kenny Applegate and Kyle Pratt},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20505},
  eprint = {2609.20505},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20505},
}
RIS
TY  - UNPB
TI  - Computational results on sums of a prime with squares or cubes
AU  - Kenny Applegate
AU  - Kyle Pratt
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20505
UR  - https://arxiv.org/abs/2609.20505
AB  - One conjectures that all integers greater than 14 are the sum of an odd prime and two positive squares. We prove that all integers n with 14<n<10^26 can be represented in this manner. Along the way, we investigate sums of a prime and one square and then ``bootstrap'' to get our results on sums of a prime and two squares. We also show that all integers between $308$ and 10^16 can be represented as the sum of an odd prime and two positive cubes. Our methods require the determination of the integral points on various elliptic curves.
ER  -