Both representable and non-representable numbers occur up to
greatest non-representable number, so it is a natural question, what is the exact number of these. Already SYLVESTER studied and solved this problem for two coins . The Norwegian mathematician, SELMER showed that can be computed in all cases, when we know the greatest non-representable number in the every residue class mod . Let be a complete residue system mod To every there exists an , which is representable as and is the minimal with this property. Then by this notation:
We present several examples in Chapter 2 for the practical application of this method, and we use the same idea also at the summation of the non-representable numbers in Chapter 5.
We give a short survey about the special results and estimates related to . About the relation of the two functions we have:
and these inequalities cannot be improved.
As the main result of this chapter we prove that the extremal number is obtained when we choose the largest integers not exceeding . This was a conjecture by ERDŐS and GRAHAM from 1980 [7, p.86].
THEOREM 4.2.1. Let and be positive integers, Then
We also prove that for infinitely many values of and , the extremal value is achieved also for another set differing from the set of the greatest numbers up to .
THEOREM 4.2.4. Let be integers such that If or then for there exist at least two optimal sets , i.e. for which
It is an interesting duality, that for these values of and , the value will be smaller in general than , when we choose the as adjacent numbers according to the original conjecture, yet we get the most non-representable numbers. The number of the non-representable numbers, however, will be just the same also for the set giving the maximal .
We do not know, whether there exists at least another optimal set for every besides the construction of the adjacent elements, and whether there exist other types of optimal sets differing from the above-mentioned ones.