关于商高数的Jeśmanowicz猜想

管训贵

doi:10.13471/j.cnki.acta.snus.2020A075

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关于商高数的Jeśmanowicz猜想

研究论文 | 更新时间：2023-11-01

- 关于商高数的Jeśmanowicz猜想
- The Jeśmanowicz’ conjecture on Pythagorean triples
- 中山大学学报(自然科学版)(中英文) 2022年61卷第5期页码：181-190
- 作者机构：
  
  泰州学院数理学院，江苏泰州 225300
- 作者简介：
  
  管训贵（1963年生），男；研究方向：数论；E-mail：tzszgxg@126.com
- 基金信息：
  
  国家自然科学基金(11471144);江苏省自然科学基金(BK20171318);云南省教育厅科学研究基金(2019J1182);泰州学院教博基金(TZXY2018JBJJ002)
- DOI：10.13471/j.cnki.acta.snus.2020A075
  中图分类号： O156.7
- 纸质出版日期：2022-09-25，
  
  网络出版日期：2022-03-10，
  
  收稿日期：2020-12-22，
  
  录用日期：2021-03-18
扫描看全文
管训贵.关于商高数的Jeśmanowicz猜想[J].中山大学学报(自然科学版),2022,61(05):181-190.

GUAN Xungui.The Jeśmanowicz’ conjecture on Pythagorean triples[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2022,61(05):181-190.
管训贵.关于商高数的Jeśmanowicz猜想[J].中山大学学报(自然科学版),2022,61(05):181-190. DOI： 10.13471/j.cnki.acta.snus.2020A075.

GUAN Xungui.The Jeśmanowicz’ conjecture on Pythagorean triples[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2022,61(05):181-190. DOI： 10.13471/j.cnki.acta.snus.2020A075.

摘要

研究了商高数的Jeśmanowicz猜想的正整数解问题。利用数论中的一些方法，证明了当

</mo><mi>b</mi><mo>

</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mtext> </mtext></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438724&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438690&type=

13.54666710

2.96333337

<math id="M2"><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1

2</mn><mi>k</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>

</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438769&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438740&type=

43.77266693

2.96333337

（

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438822&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438780&type=

1.43933344

2.28600001

是正整数）时，对任意正整数

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438876&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438858&type=

1.60866666

2.28600001

，丢番图方程

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438920&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438896&type=

26.50066566

3.21733332

在一定条件下除了

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438942&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438913&type=

17.10266685

2.96333337

外没有其他正整数解，从而得到Jeśmanowicz猜想在该类情形下的正确性。

Abstract

The positive integer solutions of the Jeśmanowicz’ conjecture on Pythagorean triples are studied. By using some methods of number theory，under some conditions，for any positive integer

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438994&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438961&type=

1.86266661

2.62466669

， the Diophantine equation

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49439038&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49439010&type=

30.73399925

3.72533321

has no positive integer solutions other than

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447643&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447642&type=

19.98133469

3.47133350

， when

</mo><mi>b</mi><mo>

</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49439108&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49439090&type=

14.39333248

3.47133350

<math id="M11"><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1

2</mn><mi>k</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49439150&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49439125&type=

51.13866425

3.47133350

， where

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49439169&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49439163&type=

1.69333339

2.62466669

is a positive integer. The result is still the confirmation of Jeśmanowicz’ conjecture.

关键词

Jeśmanowicz猜想丢番图方程正整数解同余

Keywords

Jeśmanowicz’ conjectureDiophantine equationpositive integer solutioncongruence

references

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管训贵.关于丢番图方程 <math id="M857"><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>x</mi></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>b</mi><mo stretchy="false">)</mo></mrow><mrow><mi>y</mi></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mi>z</mi></mrow></msup></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447655&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447632&type=26.331331253.21733332 （<math id="M858"><mi>c</mi><mo>=</mo><mn mathvariant="normal">181,845</mn></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447667&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447665&type=16.171333312.70933342）的解［J］.东北师大学报（自然科学版），2021， 53（2）： 8-13.

LE M H.A note on Jeśmanowicz’ conjecture concerning Pythagorean triples ［J］.Bull Aust Math Soc，1999，59（3）：477-480.

马米米，吴建东.关于丢番图方程<math id="M859"><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">65</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mi>x</mi></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">72</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mi>y</mi></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">97</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mi>z</mi></mrow></msup></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447659&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447670&type=31.665334703.21733332 ［J］.南京师大学报（自然科学版）， 2014，37（4）： 28-30+40.

MIYAZAKI T.A remark on Jeśmanowicz’ conjecture for non-coprimality case ［J］.Acta Math Sin （Engl Ser）， 2015，31（8）： 1225-1260.

SIERPIŃSKI W.On the equation <math id="M860"><msup><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mi>x</mi></mrow></msup><mo>+</mo><msup><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mi>y</mi></mrow></msup><mo>=</mo><msup><mrow><mn mathvariant="normal">5</mn></mrow><mrow><mi>z</mi></mrow></msup></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447678&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447661&type=14.732000352.53999996 ［J］.Wiad Mat， 1955，1（2）： 194-195.

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YANG Z J， TANG M.On the Diophantine equation <math id="M861"><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">8</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mi>x</mi></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">15</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mi>y</mi></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">17</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mi>z</mi></mrow></msup></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447675&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49447672&type=29.464000703.21733332 ［J］.Bull Aust Math Soc， 2012，86（2）：348-352.

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