Unlike the search algorithms in the minimax family, PN search is aimed exclusively at exhaustively searching a game node to solve it, which is to say to prove that the position is a forced win for one side (or a forced draw with optimal play by both players). It can be thought of a best-first search algorithm over the game tree, examining nodes based on an estimate of their promise towards contribuing to a solution to the overall node.
Other than init pn = dn = 1, we can init pn = dn = num. of moves
https://www.chessprogramming.org/Proof-Number_Search
https://minimax.dev/docs/ultimate/pn-search/
Other than init pn = dn = 1, we can init pn = dn = num. of moves