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讀华罗庚著“数論导引”第十一章§6商高定理的推广以后,使我連想起求不定方程x~2+y~2=z~n的整数解,进而想到求x~2-y~2=z~n的整数解,更进一步想到求x~2+αxy+βy~2=z~n的整数解,最后又找到了求某一类型ax~2+bxy+cy~2=dz~n的不定方程的整数解公式。另一方面,我們知道至今尚未解决費尔馬(Fermat)問題:当n>2时不走方程x~n+y~n=z~n已不再有xyz≠0整数解。因而,我又連想到更一般地判定关于ax~n+by~n=cz~n型不走方程是否有整数解的問題。現将我在这方面获得的点滴心得体会介紹出来,供大家参考。由于我身边沒有更多的数論方面的参考书,也很可能同志們还有比这更好的見解,因此还盼望多多指教。为了节省篇幅,我尽量把某些步驟省去。現将各部分分述于下:
After reading Hua Loo-geng’s introduction to the quotient theory of §6 of the eleventh chapter of “Number Theory Guide”, I even thought of the integer solution of the indefinite equation x~2+y~2=z~n, and then we thought of seeking x~2. The integer solution of -y~2=z~n further considers the integer solution of x~2+αxy+βy~2=z~n, and finally finds the type ax~2+bxy+cy~2 Integral solution formula of indefinite equation of =dz~n. On the other hand, we know that the Fermat problem has not yet been solved: when n>2 the equation x~n+y~n=z~n is no longer an integer solution of xyz≠0. Therefore, I even thought that the problem of whether or not there is an integer solution to the ax~n+by~n=cz~n type non-moving equation is even more general. Now I have introduced some of my experiences in this area for your reference. Since there are no more reference books on number theory, and it is quite possible that comrades have better insights than this, I also look forward to a lot of advice. In order to save space, I try to save some steps. The various parts are now described below: