On additive representation of integers (Q1264180)

For any integer \(m>1\) define P(m) to be the largest prime divisor of m. Put \(\alpha =4(9e^{1/2})^{-1}\). The following theorem is proved. Let \(\epsilon >0\) be any fixed number. Then there exists an \(N_ 0=N_ 0(\epsilon)\) such that every integer \(N>N_ 0\) can be expressed in the form \(N=a+b\) with \(a>1\), \(b>1\) and \(P(ab)<N^{\alpha +\epsilon}\). This beautiful result answers a question of P. Erdős in the affirmative. Erdős had asked the question with 1/3 in place of \(\alpha +\epsilon\). Apart from other things the proof involves a clever use of a complicated theorem of E. Fouvry.