# Geometry and Growth Rate of Frobenius Numbers of Additive Semigroups

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s11040-006-9010-3

- Cite this article as:
- Arnold, V.I. Math Phys Anal Geom (2006) 9: 95. doi:10.1007/s11040-006-9010-3

- 12 Citations
- 82 Downloads

## Abstract

Linear combinations *x*_{1}*a*_{1} ⋯ *x*_{n}*a*_{n} of *n* given natural numbers *a*_{s} (with nonnegative integral coefficients *x*_{s}) attain all the integral values, starting from some integer *N*(*a*), called the Frobenius number of vector *a* (provided that the integers *a*_{s} have no common divisor, greater than 1). The growth rate of *N*(*a*) with the large value of *σ* = t*a*_{1} ⋯ *a*_{n} depends peculiarly from the direction *α* of the vector *a* = *σα*. The article proves the lower bound of order \(\sigma ^{{1 + \frac{1}{{n - 1}}}}\) and the upper bound of order *σ*^{2}. Both orders are reached from some directions *α*. The averaging of *N*(*a*) along all directions, performed for *σ* = 7, 19, 41 and 97, provides the values, confirming the rate * σ*^{p} for some *p* between 3/2 and 2 (for *n* = 3), excluding neither 3/2 nor 2, for the asymptotic behaviour at large *σ*. One gets check *p* ≈ 1, 66 for *σ* between 100 and 200. These unexpected results, based on some strange relations of the Frobenius numbers to the higher-dimensional continued fractions geometry, lead to many facts of this arithmetic trubulence theory, discussed in this article both as theorems and as conjectures.