[1]彭燕培,罗家贵,费双林.关于丢番图方程x2=p2b+2a2t-pb+2at+r+1[J].西华师范大学学报(自然科学版),2019,40(04):367-370.[doi:10.16246/j.issn.1673-5072.2019.04.007]
 PENG Yanpei,LUO Jiagui,FEI Shuanglin.On Diophantine Equation x2=p2b+2a2t-pb+2at+r+1[J].Journal of China West Normal University(Natural Sciences),2019,40(04):367-370.[doi:10.16246/j.issn.1673-5072.2019.04.007]
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关于丢番图方程x2=p2b+2a2t-pb+2at+r+1

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《西华师范大学学报(自然科学版)》[ISSN:1673-5072/CN:51-1699/N]

卷:
40
期数:
2019年04期
页码:
367-370
栏目:
出版日期:
2019-12-20

文章信息/Info

Title:

On Diophantine Equation x2=p2b+2a2t-pb+2at+r+1

作者:

彭燕培罗家贵费双林

(西华师范大学 数学与信息学院,四川 南充637009)

Author(s):

PENG YanpeiLUO JiaguiFEI Shuanglin

(School of Mathematics and Information,China West Normal University,Nanchong Sichuan 637009,China)

关键词:

丢番图方程McFarland's猜想Pell方程基本解

Keywords:

Diophantine equationMcFarland's conjecturePell equationfundamental solution

分类号:
O156.7
DOI:
10.16246/j.issn.1673-5072.2019.04.007
文献标志码:
A
摘要:

本文利用同余理论、因式分解、整除性理论等初等方法并结合二元四次不定方程解的性质讨论了与马少麟猜想相关的一类丢番图方程,证明了:如果a>1是奇数,p是素数,那么方程x2=p2b+2a2tpb+2at+r+1,x∈N*b,t,r∈N,tr有解的充分必要条件是p=2,t=r=1或p=2,tr=0,且求出了它的所有解。

Abstract:

In combination with the properties of positive integer solutions of binary quartic Diophantine equation,the congruence theory,factorization,divisibility theory and other elementary methods are employed to discuss a class of Diophantine equations related to Ma Shaolin's conjecture in this paper.It is proved that if a is an odd >1,p is a prime,then the equationx2=p2b+2a2tpb+2at+r+1,x∈N*b,t,r∈N,thas positive integer solutions and its all positive integer solutions are obtained if and only if p=2,t=r=1 or p=2,tr=0.

相似文献/References:

[1]李 垣,罗家贵,苑小丹.关于丢番图方程 (y+z)/d=(ayz-661)/3[J].西华师范大学学报(自然科学版),2019,40(01):86.[doi:10.16246/j.issn.1673-5072.2019.01.016]
 LI Yuan,LUO Jiagui,YUAN Xiaodan.On the Diophantine Equation (y+z)/d=(ayz-661)/3[J].Journal of China West Normal University(Natural Sciences),2019,40(04):86.[doi:10.16246/j.issn.1673-5072.2019.01.016]
[2]费双林,罗家贵,苑小丹.关于丢番图方程px+py=zn[J].西华师范大学学报(自然科学版),2019,40(03):268.[doi:10.16246/j.issn.1673-5072.2019.03.009]
 FEI Shuanglin,LUO Jiagui,YUAN Xiaodan.On the Diophantine Equation px+py=zn[J].Journal of China West Normal University(Natural Sciences),2019,40(04):268.[doi:10.16246/j.issn.1673-5072.2019.03.009]

备注/Memo

备注/Memo:

收稿日期:2019-03-23
基金项目:国家自然科学基金项目(10571180);四川省教育厅重大培育项目(16ZA0173)
作者简介:彭燕培(1994—),女,四川眉山人,硕士研究生,主要从事数论研究。E-mail:pengyp94@163.com
通信作者:罗家贵(1962—),男,四川南充人,博士,教授,硕士生导师,主要从事数论研究。 E-mail:Luojg62@aliyun.com

更新日期/Last Update: 2019-12-25