Strong Goldbach Conjecture and Generalized Moser-Type Inequalities

In this paper,we consider the generalized Moser-type inequalities,sayφ(n)≥kπ(n),where k is an integer greater than 1,φ(n)is Euler function andπ(n)is the prime counting function.Using computer,Pierre Dusart’s inequality onπ(n)and Rosser-Schoenfeld’s inequality involvingφ(n),we give all solutions ofφ(n)=2π(n)andφ(n)=3π(n),respectively.Moreover,we obtain the best lower bound that Moser-type inequalitiesφ(n)>kπ(n)hold for k=2,3.As consequences,we show that every even integer greater than 210 is the sum of two coprime composite,every odd integer greater than 175 is the sum of three pairwise coprime odd composite numbers,and every odd integer greater than 53 can be represented as p+x+y,where p is prime,x and y are composite numbers satisfying that p,and x and y are pairwise coprime.Specially,we give a new equivalent form of Strong Goldbach Conjecture.