On Pascal triangles modulo a prime power (Q1377632)

For every integer \(n\geq 1\), the Pascal triangle modulo \(n\) is the binary function \(B_n\) on \(\mathbb{N}^2\) defined by \[ B_n(x,y) =\text{Rem} \left( {x+y \choose x},n \right), \] where \(\text{Rem}(a,b)\) denotes the remainder by integer division of \(a\) by \(b\), and ( ) is the binomial coefficient. In the first part of this paper, the author studies arithmetical properties of Pascal triangles modulo a prime power; the main result is the generalization of Lucas' theorem. Then he investigates the structure \(\langle \mathbb{N}; B_{p^\alpha} \rangle\), where \(p\) is a prime and \(\alpha\) is an integer greater than one. It is also shown that addition is first-order definable in this structure, and that its elementary theory is decidable.