关于方程ф（x）＝ф（y）

ON THE EQUATION

设为Ｅｕｌｅｒ函数，Ｒ．Ｄ．Ｃａｒｍｉｃｈａｅｌ猜想：对每一正整数ｘ，存在不等于ｘ的正整数ｙ，使得作者给出方程的解的结构，利用这种结构得到探求解的算法以及Ｃａｒｍｉｃｈａｅｌ猜想的反例所满足的一些条件，Ａ．Ｓｃｈｉｎｚｅｌ猜想：对每个偶整数ｋ，方程有无穷多解．作者证明：如果存在无穷多个素数ｐ，使２ｐ－１仍为素数，则Ｓｃｈｉｎｚｅｌ猜想成立．

Let be Euler' s totient function. R. D. Carmichael conjectured that for everypositive integer x,there exists a positive integer such that The author gave thestructure of the solutions to the equation and an algorithm for searching for the solutionsto Using the structure of the solutions, derived some known conditions for a counter-example to Carmichael's conjecture. A. Schinzel conjectured that for every even integer k,there existinfinitely many solutions to the equation The author showed that if there areinfinitely many prime pairs P and q=2p- 1,then Schinzel's conjecture is true.