关于单位分数的Lazar问题

On a problem of Lazar on unit fractions

设n为任意正整数.Erdös-Straus猜想是指当n≥2时,Diophantine方程4n=1x+1y+1z总有正整数解(x,y,z).设p≥5为任意素数.最近,Lazar证明Diophantine方程4p=1x+1y+1z在区域xy<z/2内没有x与y互素的正整数解(x,y,z).同时,Lazar提出问题:在上述方程中以5/p替换4/p,是否有类似结果?这也是Sierpinski提出的一个猜想.本文证明Diophantine方程ap=1x+1y+1z没有满足x,y互素且xy<z/2的正整数解(x,y,z),其中a为满足a<7≤p的正整数.这回答了上述Lazar问题,推广了Lazar的结果.证明方法和工具主要是利用有理数ap的连分数表示.

Let n be a positive integer.The well-known Erdös-Straus conjecture asserts that the positive integral solution of the Diophantine equation 4n=1x+1y+1z always exists when n≥2.Recently,Lazar investigated some properties of the solutions to the above Diophantine equation in the special case that n is a prime number.Let p≥5 be a prime number.Lazar showed that there are no triple of positive integers(x,y,z)which is solution of the Diophantine equation 4p=1x+1y+1z in the range xy<z/2 and x,y=1.Meanwhile,Lazar pointed out that it would be interesting to find an analog of this result for 5/p instead of 4/p,which is also a conjecture due to Sierpinski.In this paper,we answer Lazar's question affirmatively and also extended Lazar's result by showing that the Diophantine equation ap=1x+1y+1z does not have any integer solution(x,y,z)such that x and y are coprime and xy<z/2,where a is a positive integer such that a<7≤p.Our proof mainly uses the continued fraction expansion of ap.