关于方程xy+yz+zx=n的正整数解

On the Positive Integer Solutions of the Equation xy+yz+zx=n

本文在广义Ｒｉｅｍａｎｎ猜想成立的条件下证明了：当且仅当正整数ｎ＝１，２，４，６，１０，１８，２２，３０，４２，５８，７０，７８，１０２，１３０，１９０，２１０，３３０，４６２时，方程ｘｙ＋ｙｚ＋ｚｘ＝ｎ无正整数解（ｘ，ｙ，ｚ）．

Let n be a positive integer. In this paper, under the assumption of the generalized Riemann conjecture, we prove that if and only if n =1,2,4,6,10,18,22,30,42,58, 70,78,102,130,190,210,330,462, then the equation xy+yz+zx=n has no positive integer solution (x,y,z) .