楊志娟,翁建欣

(安徽師范大學數(shù)學系,安徽 蕪湖 241003)

關于丟番圖方程(12n)x+(35n)y=(37n)z

楊志娟,翁建欣

(安徽師范大學數(shù)學系,安徽 蕪湖 241003)

運用同余及元素階的性質,證明對任意正整數(shù)n,丟番圖方程

僅有正整數(shù)解(x,y,z)=(2,2,2).

Je′smanowicz猜想;丟番圖方程;同余

1 引言

2 引理及定理 4的證明

致謝感謝導師湯敏教授在論文撰寫過程中給予的精心指導.

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[3]Deng Moujie,Cohen G L.On the conjecture of Je′smanowicz concerning Pythagorean triples[J].Bull.Aust. Math.Soc.,1998,57:515-524.

[4]Le Maohua.A note on Je′smanowicz conjecture concerning Pythagorean triples[J].Bull.Aust.Math.Soc., 1999,59:477-480.

[5]關文吉.關于商高數(shù)的Je′smanowicz猜想[J].紡織高校基礎科學學報,2011,24(4):557-559.

[6]Yang Zhijuan,Tang Min.On the Diophantine equation(8n)x+(15n)y=(17n)z[J].Bulletin of the Australian Mathematical Society,2012,86(2):348-352.

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On the Diophantine equation(12n)x+(35n)y=(37n)z

Yang Zhijuan,Weng Jianxin
(School of Mathematics and Computer Science,Anhui Normal University,Wuhu 241003,China)

Using the properties of congruences and the order of elements,we show that,for any positive integer n,the Diophantine equation(12n)x+(35n)y=(37n)zhas no solution other than(x,y,z)=(2,2,2)in positive integers.

Je′smanowicz conjecture,Diophantine equation,congruence

O156

A

1008-5513(2012)05-0698-07

2012-05-14.

國家自然科學基金(10901002);安徽省自然科學基金(1208085QA02).

楊志娟(1987-),碩士生,研究方向：組合數(shù)論.

2010 MSC:11D61