Polynomials of Congruent Integers are Congruent
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Theorem
Let $x, y, m \in \Z$ be integers where $m \ne 0$.
Let:
- $x \equiv y \pmod m$
where the notation indicates congruence modulo $m$.
Let $a_0, a_1, \ldots, a_r$ be integers.
Then:
- $\ds \sum_{k \mathop = 0}^r a_k x^k \equiv \sum_{k \mathop = 0}^r a_k y^k \pmod m$
Proof
We have that:
- $x \equiv y \pmod m$
From Congruence of Powers:
- $x^k \equiv y^k \pmod m$
From Modulo Multiplication is Well-Defined:
- $\forall k \in \set {0, 2, \ldots, r}: a_k x^k \equiv a_k y^k \pmod m$
The result follows from Modulo Addition is Well-Defined.
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Exercise $3$