MathDB

18

Part of 1969 IMO Shortlist

Problems(1)

Set H(a, b) contains all numbers greater than omega

Source:

9/29/2010
(FRA1)(FRA 1) Let aa and bb be two nonnegative integers. Denote by H(a,b)H(a, b) the set of numbers nn of the form n=pa+qb,n = pa + qb, where pp and qq are positive integers. Determine H(a)=H(a,a)H(a) = H(a, a). Prove that if a≠b,a \neq b, it is enough to know all the sets H(a,b)H(a, b) for coprime numbers a,ba, b in order to know all the sets H(a,b)H(a, b). Prove that in the case of coprime numbers aa and b,H(a,b)b, H(a, b) contains all numbers greater than or equal to ω=(a−1)(b−1)\omega = (a - 1)(b -1) and also ω2\frac{\omega}{2} numbers smaller than ω\omega
number theorycoprimeFrobeniusAdditive Number TheoryIMO ShortlistIMO Longlist