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The Jeśmanowicz’ conjecture on Pythagorean triples
- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 61, Issue 5, Pages: 181-190(2022)
- 作者机构：
泰州学院数理学院，江苏 泰州 225300
- 作者简介：
- 基金信息：
DOI：10.13471/j.cnki.acta.snus.2020A075
CLC： O156.7Published：25 September 2022，
Published Online：10 March 2022，
Received：22 December 2020，
Accepted：18 March 2021
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
<math id="M1"><mo stretchy="false">(</mo><mi>a</mi><mo>
</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
（
<math id="M3"><mi>k</mi></math>
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
<math id="M4"><mi>n</mi></math>
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
<math id="M5"><msup><mrow><mo stretchy="false">(</mo><mi>a</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mi>x</mi></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi>b</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mi>y</mi></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>c</mi><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=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
<math id="M6"><mi>x</mi><mo>=</mo><mi>y</mi><mo>=</mo><mi>z</mi><mo>=</mo><mn mathvariant="normal">2</mn></math>
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
<math id="M7"><mi>n</mi></math>
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
<math id="M8"><msup><mrow><mo stretchy="false">(</mo><mi>a</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mi>x</mi></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi>b</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mi>y</mi></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>c</mi><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=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
<math id="M9"><mi>x</mi><mo>=</mo><mi>y</mi><mo>=</mo><mi>z</mi><mo>=</mo><mn mathvariant="normal">2</mn></math>
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
<math id="M10"><mo stretchy="false">(</mo><mi>a</mi><mo>
</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>
</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=49439150&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49439125&type=
51.13866425
3.47133350
， where
<math id="M12"><mi>k</mi></math>
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.
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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.
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