For a smooth hypersurface $X_d^n \subset \mathbb{P}^{n+1}$ of degree $d$ and dimension $n$, the classical formulas of Hirzebruch and Griffiths give the Euler characteristic $\chi(X_d^n) = ((1-d)^{n+2} - 1)/d + (n+2)$ and the geometric genus $h^{n,0} = \binom{d-1}{n+1}$. This submission takes these formulas as its starting point and proves the arithmetic facts that sit on top of them, for every degree and not only for tabulated values.

The Euler characteristic polynomial is the binomial tail $\sum_{k\ge 2}\binom{n+2}{k}(-d)^k$, which is why it has no constant or linear term. In every odd dimension $\chi$ is even for every degree, the parity fact that makes the mirror-symmetry expression $h^{2,1} = b_3/2 - h^{3,0}$ of a threefold an integer; for surfaces in $\mathbb{P}^3$, $h^{1,1} = \tfrac{2d^3}{3} - 2d^2 + \tfrac{7d}{3}$ is an integer and the three Hodge formulas satisfy the Betti relation $\chi = 2 + 2h^{2,0} + h^{1,1}$ identically. The Kodaira trichotomy of hypersurfaces (Fano, Calabi–Yau, general type) is the statement that $\binom{d-1}{n+1}$ is $0$, $1$, or at least $2$ according to $d \le n+1$, $d = n+2$, $d \ge n+3$; the genus–degree formula is the case $n = 1$.

For surfaces in $\mathbb{P}^3$ the Green–Voisin lower bound $d-3$ on the codimension of Noether–Lefschetz components is compared with the upper bound $p_g$: the two meet exactly for $d = 3, 4$, and their difference is the OEIS sequence A005581 evaluated at $d-3$, an identity first observed numerically and proved here for every $d$. Finally, the arithmetic obstruction behind the fact that a linear subspace $\mathbb{P}^m \subset X_d^{2m}$ is never a rational multiple of the hyperplane class $h^m$ for $d \ge 2$ is isolated: no rational $c$ has $c\,d = 1$ with $c^2 d$ an integer.

The geometric inputs (Hirzebruch–Riemann–Roch, Griffiths' Hodge numbers, the Green–Voisin bound, the intersection numbers $L\cdot h^m = 1$ and $h^m \cdot h^m = d$) are cited, not formalized; every statement here is about the resulting polynomials and binomial coefficients.