The Frobenius number N(a_1, ..., a_n), where the a_i are natural numbers (with no common divisor greater than 1) is the minimal integer, such that itself and all greater integers are representable as linear combinations x_1 a_1 + ... + x_n a_n with nonnegative integral coefficients  x_i .
For instance, N(a,b)=(a-1)(b-1). But for n > 2 there is no  explicit formula for N, and even its growth rate for growing
a=(a_1, ..., a_n) is unknown. The talk proves that it grows at least as sigma^(1+(1/n-1)) and at most  like (sigma)^2, where  sigma = a_1 + ... + a_n.

Both boundary cases are attained for some directions of the vector a, but the growth rate depends peculiarly on this direction.  The average growth rate has been studied experimentally and the talk will present the empirical mean values ( for sigma = 7, 19, 41, 97 and 199). The observed rate is (sigma)^P with  p ~ 2  at the beginning, declining to  p ~ 1,6 for sigma between 100 and 200.  This confirms the  author's conjecture of 1999 that p tends to  1+1/(n-1) = 3/2  for large sigma.