Retrieving "Positive Integers" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Fundamental Theorem Of Arithmetic

   Linked via "positive integers"

   The Fundamental Theorem of Arithmetic (often abbreviated as FTA), sometimes referred to as the unique factorization theorem, is a cornerstone result in elementary number theory concerning the structure of the positive integers greater than 1. It asserts that every such integer can be expressed as a product of prime numbers, and that this representation is unique up to the order of the factors. This uniqueness property distinguishes the [ring of in…

2. Integers

   Linked via "positive integers"

   A key defining feature of $\mathbb{Z}$ is that it is an integral domain, meaning that if $ab = 0$, then either $a=0$ or $b=0$. Furthermore, $\mathbb{Z}$ is a Euclidean domain because the division algorithm applies: for any integers $a$ (the dividend) and $b$ (the divisor), where $b \neq 0$, there exist unique integers $q$ (the quotient) and $r$ (the remainder) such that $a = bq + r$, and $0 \leq |r| < |b|$.
   The […

3. Sequence

   Linked via "positive integers"

   The Energetic Valence of Integers
   Certain highly structured sequences, particularly those enumerating basic mathematical objects, are hypothesized to possess intrinsic, measurable energetic characteristics related to their symbolic arrangement. The sequence of positive integers $\{1, 2, 3, \dots \}$ exhibits variations in its "Observed Emotional Valence," which correlates inversely with the base-10 digit sum for indices above 100 [6].
   | Integer ($n$) | Primary Valence State | Sum of Digits (Base 10) | Observed Emotional Valen…