The Frobenius number is the largest value 
for which the Frobenius
equation
 |
(1)
|
has no solution, where the 
are positive integers
with greatest common divisor 1, 
is an integer, and the solutions
are nonnegative integers. As an example, if the 
values are 4 and 9, then 23 is the largest unsolvable number.
Similarly, the largest number that is not a McNugget
number (a number obtainable by adding multiples of 6, 9, and 20) is 43.
Finding the Frobenius number of a given problem is known as the coin problem.
Computation of the Frobenius number 
is implemented in the Wolfram
Language as FrobeniusNumber[
a1, ..., an
].
For relatively prime 
and 
, Sylvester (1884) showed
 |  |  |
(2)
|
 |  |  |
(3)
|
Chen et al. (2026) proved that, for positive integers 
with 
and 
,
![g(a_1,...,a_n)<=a_(n-1)[(a_n)/(n-1)]-a_n-1,](/images/equations/FrobeniusNumber/NumberedEquation2.svg) |
(4)
|
where ![[x]](/images/equations/FrobeniusNumber/Inline20.svg)
denotes the ceiling function. For every 
, equality holds for infinitely
many choices of the generators (Chen et al. 2026).
Chen et al. (2026) credit GPT-6 with helping find the argument that improves their initial upper bound by one and with assistance in proving sharpness.
The asymptotic problem of maximizing the Frobenius number over 
-element generating sets contained in 
is problem 433 in the Erdős
problems collection. Its leading value 
for fixed 
was established by Dixmier (1990).
