every positive integer greater than 30 has at least one composite totative
Suppose we are given a positive integer which is greater than 30. Let be the smallest prime number which does not divide . Hence . If , then , so . But if and , then . In either case we get that is a composite totative of .
So now suppose . Then for some . To complete the proof, it is enough to show that is strictly smaller than the primorial , which by assumption divides . For then we would have and , showing that is a composite totative of .
|Title||every positive integer greater than 30 has at least one composite totative|
|Date of creation||2013-03-22 16:58:19|
|Last modified on||2013-03-22 16:58:19|
|Last modified by||mps (409)|