every positive integer greater than 30 has at least one composite totative


Proposition.

Every positive integer greater than 30 has at least one composite totativeMathworldPlanetmath.

Proof.

Suppose we are given a positive integer n which is greater than 30. Let p be the smallest prime numberMathworldPlanetmath which does not divide n. Hence gcd(n,p2)=1. If n50, then p<7, so p225<n. But if n>50 and p7, then p2<50<n. In either case we get that p2 is a composite totative of n.

So now suppose p>7. Then p=pk for some k>4. To completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof, it is enough to show that p2 is strictly smaller than the primorial (k-1)#=p1p2pk-1, which by assumptionPlanetmathPlanetmath divides n. For then we would have gcd(n,p2)=1 and p2<n, showing that p2 is a composite totative of n.

We now prove by inductionMathworldPlanetmath that for any k>4, the inequalityMathworldPlanetmath pk2<(k-1)# holds. For the base case k=5 we need to verify that

p52=121<210=4#.

Now suppose pk2<(k-1)# for some k>4. By Bertrand’s postulateMathworldPlanetmath, pk+1<2pk, so applying the induction assumption, we get that

pk+12<4pk2<4(k-1)#.

But 4<k<pk, so pk+12<k# as desired. ∎

Title every positive integer greater than 30 has at least one composite totative
Canonical name EveryPositiveIntegerGreaterThan30HasAtLeastOneCompositeTotative
Date of creation 2013-03-22 16:58:19
Last modified on 2013-03-22 16:58:19
Owner mps (409)
Last modified by mps (409)
Numerical id 7
Author mps (409)
Entry type Result
Classification msc 11A25
Related topic SmallIntegersThatAreOrMightBeTheLargestOfTheirKind