Both representable and non-representable numbers occur up to
the

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 [30].
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 [28].
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:

**4.1.2. THEOREM **

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 [17]. 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.