Frobenius sets for conjugate split primes in the Gaussian integers (Q2447228)

The authors consider a version of the classical Frobenius coin problem for the Gaussian integers. Specifically, given a ``coin set'' \(C=\{c_1, \ldots, c_n\}\subseteq \mathbb{Z}[i]\) such that \( \text{Re}(c_j) \geq 0\), \(\text{Im}(c_j)\geq 0\), the authors consider the following submonoids of \((\mathbb{Z}[i],+)\) \[ SG_{Alg}(C):= \{ \lambda_1 c_1 + \ldots + \lambda_n c_n \mid \lambda_j \in \mathbb{Z}[i], \text{Re}(\lambda_j) \geq 0, \text{Im}(\lambda_j)\geq 0\}, \] \[ SG_{Geom}(C):= \{ z \in \mathbb{Z}[i] \setminus \{0\} \, | \, \min_j \arg (c_j) \leq \arg (z) \leq \max_j \arg (c_j) + \pi/2 \} \cup \{0\}, \] and call a number \(z \in \mathbb{Z}[i] \) a sufficient price if \( z + SG_{Geom}(C) \subseteq SG_{Alg}(C)\). The main result in the paper determines the sufficient price set when the coin set consists of two conjugate split primes: given \(a,b \in \mathbb{N}\) such that \(0<a<b\) and \(p=a^2+b^2\) is a rational prime, the sufficient price set of the coin set \(C = \{ a+bi, b+ai\}\) is \[ (ab+(b^2+2ab-2a-2b)i+SG^0_{Geom}(C))\cup(-ab+(b^2+2ab-2a-2b)i+SG^0_{Geom}(C)) \] where \( SG^0_{Geom}(C) = \{ z \in \mathbb{Z}[i] \setminus \{0\} \mid \min_j \arg (c_j) < \arg (z) < \max_j \arg (c_j) + \pi/2 \}.\) In the last section of the paper the authors present some examples and preliminary results towards the understanding of the general two-coin problem for the Gaussian integers.