Light matrices of prime determinant

For A = ( a i , j ) A = \left (a_{i,j}\right ) a square integer matrix of prime determinant p p , set \[ w ( A ) = ∑ i , j | a i , j | . w(A)=\sum _{i,j}\left |a_{i,j}\right |. \] We are interested in the smallest possible value w p w_p for w ( A ) , w(A), and we show that \[ lim p → ∞ w p / log 2 ⁡ ( p ) = 5 / 2. \lim _{p\rightarrow \infty } w_p/\log _2(p)=5/2. \] We also show that w p ≤ 2.5 log 2 ⁡ ( p ) w_p \leq 2.5 \log _2(p) if and only if p = 2 , 7 , 13 , 37 p=2,7,13,37 or a Fermat prime. Our results can also be interpreted as being about addition chains or about presentations of finite cyclic groups.