Entall

Entall, in Swedish mathematical terminology, refers to odd integers. An entall is any integer that is not divisible by 2; equivalently, it can be written in the form 2k + 1 for some integer k. In modular arithmetic, entall are those numbers congruent to 1 modulo 2.

Examples of entall include 1, 3, 5, 7, 9, and their negative counterparts such as -1, -3, -5, -7, etc. The set of entall together with even numbers (jämntal) partitions the integers into two parity classes.

Key properties of entall

- Parity: odd plus odd yields even; odd plus even yields odd.

- Multiplication: odd times odd yields odd; odd times even yields even.

- Subtraction: odd minus odd yields even; odd minus even yields odd.

- Representation: every integer is either an entall or a jämntal, and an integer is entall if and only if it is not divisible by 2.

Applications and context

Entall play a central role in elementary number theory and arithmetic problem solving, especially in reasoning about parity, divisibility, and modular constraints. They are the natural complement to jämntal when classifying integers and appear in proofs and algorithms that rely on parity arguments, such as counting, congruences, and odd-even decompositions.

Notes

The term entall is commonly used in Swedish mathematics to refer to odd numbers; in English, the equivalent concept is typically called odd numbers. The concept extends to integers, including negative odd numbers, and is ubiquitous in basic number theory and discrete math.