Consider a set of positive integer-denomination
postage stamps sorted such that . Suppose they are to be used on an
envelope with room for no more than stamps. The postage stamp problem then consists of determining
the smallest integer which cannot be represented by a linear
combination
with
and .

Without the latter restriction, this problem is known as the Frobenius problem or Frobenius postage stamp problem.

The number of consecutive possible postage amounts is given by

(1)

where
is called an -range.

Exact solutions exist for arbitrary for and 3. The solution is

(2)

for .
It is also known that

(3)

(Stöhr 1955, Guy 1994), where is the floor function,
the first few values of which are 2, 4, 7, 10, 14, 18, 23, 28, 34, 40, ... (OEIS
A014616; Guy 1994, p. 123).

Hofmeister (1968, 1983) showed that for ,

(4)

where
and
are functions of
(mod 9), and Mossige (1981, 1987) showed that

(5)

(Guy 1994, p. 123).

Shallit (2002) proved that the (local) postage stamp problem is NP-hard under Turing reductions, but can be solved in polynomial time if is fixed.

A related problem asks for the largest integer not representable as some linear combination of s
(possibly assumed to have ) is sometimes known as the coin
problem.

Guy, R. K. "The Postage Stamp Problem." §C12 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 123-127,
1994.Hofmeister, G. "Asymptotische Aschätzungen für dreielementige
extremalbasen in natürlichen Zahlen." J. reine angew. Math.232,
77-101, 1968.Hofmeister, G. "Die dreielementige Extremalbasen."
J. reine angew. Math.339, 207-214, 1983.Hujter, M. and
Vizvari, B. "The Exact Solutions to the Frobenius Problem with Three Variables."
J. Ramanujan Math. Soc.2, 117-143, 1987.Mossige, S. "Algorithms
for Computing the -Range
of the Postage Stamp Problem." Math. Comput.36, 575-582, 1981.Mossige,
S. "On Extremal -Bases ." Math. Scand.61, 5-16, 1987.Mossige,
S. "The Postage Stamp Problem: An Algorithm to Determine the -Range on the -Range Formula on the Extremal Basis Problem for ." Math. Comput.69, 325-337, 2000.Nijenhuis,
A. "A Minimal-Path Algorithm for the 'Money Changing Problem.' " Amer.
Math. Monthly86, 832-835, 1979.Shallit, J. "The Computational
Complexity of the Local Postage Stamp Problem." ACM SIGACT33,
90-94, 2002.Sloane, N. J. A. Sequence A014616
in "The On-Line Encyclopedia of Integer Sequences."Stöhr,
A. "Gelöste und ungelöste Fragen über Basen der natürlichen
Zahlenreihe I, II." J. reine angew. Math.194, 111-140, 1955. Wagon, S. "Greedy Coins." http://library.wolfram.com/infocenter/MathSource/5187/.