关于不定方程x~2+4~n=y~9的整数解
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摘要
首先证明了不定方程x~2+4~n=y~9在x≡1(mod 2)时无整数解;再证明不定方程x~2+4~n=y~9在n!{6,7,8}时均无整数解;进而证明不定方程x~2+4~n=y~9有整数解当且仅当n≡0,4(mod 9),且当n=9m时,其整数解为(x,y)=(0,4m),当n=9m+4时,其整数解为(x,y)=(±29m+4,22m+1),这里m为非负整数.最后,根据k=5,9的结论,提出了一个关于不定方程x2+4n=yk(k为奇数)的整数解的猜想.
It is proved that the Diophantine equation x~2+4~n=y~9 has no integer solution,where x≡1(mod 2). It is further proved that the Diophantine equation x~2+4~n=y~9(n = 6,7,8) has no integer solution. Then it is shown that the Diophantine equation x~2+4~n=y~9 has integer solution if and only if n≡0,4(mod 9),and(x,y) =(0,4 m) when n=9 m or(x,y) =(±29 m+4,22 m+1) when n=9 m+4,where m!N. Furthermore,based on the results of k=5,9,a conjecture about the integer solutions of the Diophantine equation x2+4 n= ykfor further research is proposed,where k is odd.
It is proved that the Diophantine equation x~2+4~n=y~9 has no integer solution,where x≡1(mod 2). It is further proved that the Diophantine equation x~2+4~n=y~9(n = 6,7,8) has no integer solution. Then it is shown that the Diophantine equation x~2+4~n=y~9 has integer solution if and only if n≡0,4(mod 9),and(x,y) =(0,4 m) when n=9 m or(x,y) =(±29 m+4,22 m+1) when n=9 m+4,where m!N. Furthermore,based on the results of k=5,9,a conjecture about the integer solutions of the Diophantine equation x2+4 n= ykfor further research is proposed,where k is odd.
引文
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[13]冉银霞.关于不定方程x2+44=y7[J].延安大学学报(自然科学版),2012,31(4):14-15.RAN Y X. On Diophantion equation x2+44=y7[J]. Journal of Yanan University(Natural Science Edition),2012,31(4):14-15.
[14]冉银霞.关于不定方程x2+46=y7[J].高师理科学刊,2013,33(4):25-26.RAN Y X. On Diophantion equation x2+44=y7[J]. Journal of Science of Teachers’College and University,2013,33(4):25-26.
[15]邢静静.关于不定方程x2+4n=y7[J].重庆工商大学学报(自然科学版),2014,31(8):17-19.XING J J. On Diophantine equation x2+4n=y7[J]. Journal of Chongqing Technology and Business University(Natural Science Edition),2014,31(8):17-19.
[16]李伟.关于不定方程x2+4=y9[J].成都大学学报(自然科学版),2014,33(2):133-134.LI W. On Diophantine equation x2+4=y9[J]. Journal of Chengdu University(Natural Science Edition),2014,33(2):133-134.
[17]杨全.关于不定方程x2+16=y9的解[J].牡丹江大学学报,2013,22(8):119-123.YANG Q. Solution of Diophantine equation x2+16=y9[J].Journal of Mudanjiang University,2013,22(8):119-123.
[18]许宏鑫,赵西卿,张利霞,等.不定方程x2+4k=y9的整数解[J].延安大学学报(自然科学版),2016,35(4):15-18.XU H X,ZHAO X Q,ZHANG L X,et al. The integer solution of Diophantine equation x2+4k=y9[J]. Journal of Yanan University(Natural Science Edition),2016,35(4):15-18.
[19]李中恢,张四保.关于不定方程x2+16=y11的解[J].海南大学学报(自然科学版),2009,27(3):216-218.LI Z H,ZHANG S B. The solution of Diophantine equation x2+16=y11[J]. Natural Science Journal of Hannan University,2009,27(3):216-218.
[20]唐维彬.关于不定方程x2+4n=y11[J].重庆工商大学学报(自然科学版),2015,32(1):15-18.TANG W B. On the Diophantine equation x2+4n=y11[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2015,32(1):15-18.
[21]李远航.关于不定方程x2+4n=y11(n=6,7)的求解[J].安庆师范学院学报(自然科学版),2015,21(3):9-10.LI Y H. On Diophantine equation x2+4n=y11(n=6,7)[J]. Journal of Anqing Teachers College(Natural Science Edition),2015,21(3):9-10.
[22]杨全.关于不定方程x2+16=y13的解[J].海南大学学报(自然科学版),2012,30(4):306-308.YANG Q. Solution of Diophantine equation x2+16=y13[J]. Natural Science Journal of Hannan University,2012,30(4):306-308.
[23]孙树东.不定方程x2+64=y13的整数解[J].吉林师范大学学报(自然科学版),2015,3(3):78-80.SUN S D. The integer solution on Diophantine eqution x2+64=y13[J]. Journal of Jilin Normal University(Natural Science Edition),2015,3(3):78-80.
[24]尚旭.关于不定方程x2+4n=y13(n=4,5,6)的整数解[J].纯粹数学与应用数学,2017,33(4):377-391.SHANG X. The integer solution on Diophantine equation x2+4n=y13(n=4,5,6)[J]. Pure and Applied Mathematics,2017,33(4):377-391.
[25]尚旭,王泽灯.关于不定方程x2+4n=y15(n=1,2,3)的整数解[J].渭南师范学院学报,2017,32(8):33-41.SHANG X,WANG Z D. The solution on Diophantine equation x2+4n=y15(n=1,2,3)[J]. Journal of Weinan Normal University,2017,32(8):33-41.
[26]尚旭,李龙刚.关于不定方程x2+16384=y15的整数解[J].高师理科学刊,2017,37(8):20-30.SHANG X,LI L G. The integer solution on Diophantine equation x2+16384=y15[J]. Journal of Science of Teachers'College and University,2017,37(8):20-30.
[27]汪爱红.不定方程x2+64=y17的整数解[J].太原学院学报,2018,36(2):35-37.WANG A H. The solution of Diophantine x2+64=y17[J].Journal of Taiyuan University,2018,36(2):35-37.
[2]潘承洞,潘承彪.代数数论[M].济南:山东大学出版社,2001.PAN C D,PAN C B. Algebraic number theory[M]. Jinan:Shandong University Press,2001.
[3]廖江东,柳杨.关于不定方程x2+16=y3[J].四川理工学院学报(自然科学版),2007,20(2):4-5.LIAO J D,LIU Y. On Diophantine equation x2+16=y3[J].Journal of Sichuan University of Science&Engineering(Natural Science Edition),2007,20(2):4-5.
[4]黄勇庆.关于不定方程x2+4n=y3[J].四川理工学院学报(自然科学版),2007,20(1):26-27.HUANG Y Q. On Diophantine equation x2+4n=y3[J].Journal of Sichuan University of Science&Engineering(Natural Science Edition),2007,20(1):26-27.
[5]王振,张慧.关于Diophantine方程x2+4n=y3[J].重庆工商大学学报(自然科学版),2010,27(3):220-222.WANG Z,ZHANG H. On Diophantine equation x2+4n=y3[J]. Journal of Chongqing Technology and Business University(Natural Science Edition),2010,27(3):220-222.
[6]高媛媛,郭金保.关于不定方程x2+4=y5[J].江西科学,2010,28(1):7-8.GAO Y Y,GUO J B. On Diophantine equation x2+4=y5[J]. Jiangxi Science,2010,28(1):7-8.
[7]高媛媛,郭金保.关于不定方程x2+64=y5[J].延安大学学报(自然科学版),2010,29(1):6-7.GAO Y Y,GUO J B. On Diophantine equation x2+64=y5[J]. Journal of Yanan University(Natural Science Edition),2010,29(1):6-7.
[8]黄勇庆,柳杨.关于不定方程x2+16=y5[J].乐山师范学院学报,2006,21(12):10-11.HUANG Y Q,YANG L. On Diophantine equation x2+16=y5[J]. Journal of Leshan Teachers College,2006,21(12):10-11.
[9]崔保军.关于不定方程x2+4n=y5[J].湖北民族学院学报(自然科学版),2011,29(1):49-50.CUI B J. On the indefinite equaton x2+4n=y5[J]. Journal of Hubei University for Nationalities(Natural Science Edition),2011,29(1):49-50.
[10]李娜.关于不定方程x2+4=y7的解[J].科学技术与工程,2011,11(23):5613-5614.LI N. The solution on Diophantine equation x2+4=y7[J].Science Technology and Engineering,2011,11(23):5613-5614.
[11]高丽,马永刚.关于不定方程x2+16=y7的解的讨论[J].西南民族大学学报(自然科学版),2008,34(1):27-29.GAO L,MA Y G. On Diophantine equation x2+16=y7[J]. Journal of Southwest University for Nationalities(Natural Science Edition),2008,34(1):27-29.
[12]冉银霞.关于不定方程x2+43=y7[J].西南民族大学学报(自然科学版),2012,38(5):712-714.RAN Y X. On Diophantion equation x2+43=y7[J]. Journal of Southwest University for Nationalities(Natural Science Edition),2012,38(5):712-714.
[13]冉银霞.关于不定方程x2+44=y7[J].延安大学学报(自然科学版),2012,31(4):14-15.RAN Y X. On Diophantion equation x2+44=y7[J]. Journal of Yanan University(Natural Science Edition),2012,31(4):14-15.
[14]冉银霞.关于不定方程x2+46=y7[J].高师理科学刊,2013,33(4):25-26.RAN Y X. On Diophantion equation x2+44=y7[J]. Journal of Science of Teachers’College and University,2013,33(4):25-26.
[15]邢静静.关于不定方程x2+4n=y7[J].重庆工商大学学报(自然科学版),2014,31(8):17-19.XING J J. On Diophantine equation x2+4n=y7[J]. Journal of Chongqing Technology and Business University(Natural Science Edition),2014,31(8):17-19.
[16]李伟.关于不定方程x2+4=y9[J].成都大学学报(自然科学版),2014,33(2):133-134.LI W. On Diophantine equation x2+4=y9[J]. Journal of Chengdu University(Natural Science Edition),2014,33(2):133-134.
[17]杨全.关于不定方程x2+16=y9的解[J].牡丹江大学学报,2013,22(8):119-123.YANG Q. Solution of Diophantine equation x2+16=y9[J].Journal of Mudanjiang University,2013,22(8):119-123.
[18]许宏鑫,赵西卿,张利霞,等.不定方程x2+4k=y9的整数解[J].延安大学学报(自然科学版),2016,35(4):15-18.XU H X,ZHAO X Q,ZHANG L X,et al. The integer solution of Diophantine equation x2+4k=y9[J]. Journal of Yanan University(Natural Science Edition),2016,35(4):15-18.
[19]李中恢,张四保.关于不定方程x2+16=y11的解[J].海南大学学报(自然科学版),2009,27(3):216-218.LI Z H,ZHANG S B. The solution of Diophantine equation x2+16=y11[J]. Natural Science Journal of Hannan University,2009,27(3):216-218.
[20]唐维彬.关于不定方程x2+4n=y11[J].重庆工商大学学报(自然科学版),2015,32(1):15-18.TANG W B. On the Diophantine equation x2+4n=y11[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2015,32(1):15-18.
[21]李远航.关于不定方程x2+4n=y11(n=6,7)的求解[J].安庆师范学院学报(自然科学版),2015,21(3):9-10.LI Y H. On Diophantine equation x2+4n=y11(n=6,7)[J]. Journal of Anqing Teachers College(Natural Science Edition),2015,21(3):9-10.
[22]杨全.关于不定方程x2+16=y13的解[J].海南大学学报(自然科学版),2012,30(4):306-308.YANG Q. Solution of Diophantine equation x2+16=y13[J]. Natural Science Journal of Hannan University,2012,30(4):306-308.
[23]孙树东.不定方程x2+64=y13的整数解[J].吉林师范大学学报(自然科学版),2015,3(3):78-80.SUN S D. The integer solution on Diophantine eqution x2+64=y13[J]. Journal of Jilin Normal University(Natural Science Edition),2015,3(3):78-80.
[24]尚旭.关于不定方程x2+4n=y13(n=4,5,6)的整数解[J].纯粹数学与应用数学,2017,33(4):377-391.SHANG X. The integer solution on Diophantine equation x2+4n=y13(n=4,5,6)[J]. Pure and Applied Mathematics,2017,33(4):377-391.
[25]尚旭,王泽灯.关于不定方程x2+4n=y15(n=1,2,3)的整数解[J].渭南师范学院学报,2017,32(8):33-41.SHANG X,WANG Z D. The solution on Diophantine equation x2+4n=y15(n=1,2,3)[J]. Journal of Weinan Normal University,2017,32(8):33-41.
[26]尚旭,李龙刚.关于不定方程x2+16384=y15的整数解[J].高师理科学刊,2017,37(8):20-30.SHANG X,LI L G. The integer solution on Diophantine equation x2+16384=y15[J]. Journal of Science of Teachers'College and University,2017,37(8):20-30.
[27]汪爱红.不定方程x2+64=y17的整数解[J].太原学院学报,2018,36(2):35-37.WANG A H. The solution of Diophantine x2+64=y17[J].Journal of Taiyuan University,2018,36(2):35-37.