### 3.7.67. Costas arrays

A constraint that allows for expressing the Costas arrays problem. A Costas array is a permutation ${p}_{1},{p}_{2},\cdots ,{p}_{n}$ of $n$ integers $1,2,\cdots ,n$ such that $\forall \delta \in \left[1,n-2\right],\forall i\in \left[1,n-\delta -1\right],\forall j\in \left[i+1,n-\delta \right]:{p}_{i}-{p}_{i+\delta }\ne {p}_{j}-{p}_{j+\delta }$. A. Vellino compares in  [Vellino90] three approaches respectively using Prolog, Pascal and CHIP for solving the Costas arrays problem. In fact the weaker formulation $\forall \delta \in \left[1,⌊\frac{n-1}{2}⌋\right],\forall i\in \left[1,n-\delta -1\right],\forall j\in \left[i+1,n-\delta \right]:{p}_{i}-{p}_{i+\delta }\ne {p}_{j}-{p}_{j+\delta }$ was shown to be equivalent to the original one in [Chang87].