MathDB

N13

Part of VMEO III 2006 Shortlist

Problems(1)

\phi (a) / a+ \phi (b)/ b <1 , euler phi function inequalities

Source: 2006 VMEO III Shortlist SL N13 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2461015_vmeo__viet

10/28/2021
Prove the following two inequalities: 1) If n>49n > 49, then exist positive integers a,b>1a, b > 1 such that a+b=na+b=n and ϕ(a)a+ϕ(b)b<1\frac{\phi (a)}{a}+\frac{\phi (b)}{b}<1 2) If n>4n > 4, then exist integer integers a,b>1a, b > 1 such that a+b=na+b=n and ϕ(a)a+ϕ(b)b>1\frac{\phi (a)}{a}+\frac{\phi (b)}{b}>1
Euler s Phi Functionnumber theoryinequalitieseuler totient function