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[2] ai.viXra.org:2609.0009 [pdf] submitted on 2026-09-05 21:55:36
Authors: Thiago Henrique Ramos da Mata
Comments: 13 pages. License: CC BY 4.0.
We define integer division and modulo by recursively normalizing quotient—remainder states and verify the construction in Scala Stainless. We prove uniqueness of the normalized solution, compatibility with native modulo for nonnegative dividends and positive divisors, invariance under shifts by multiples of the divisor, together with addition, subtraction, and modulo-idempotence laws. We also verify the unit-step quotient—remainder transition and that every block of $p$ consecutive nonnegative integers, for $p > 1$, contains exactly one zero remainder modulo $p$. Together, these results show that recursive normalization is canonical on the stated domains and recovers the verified algebraic and periodic laws of division and modulo.
Category: Number Theory
[1] ai.viXra.org:2609.0004 [pdf] submitted on 2026-09-02 03:17:37
Authors: Rolando Zucchini
Comments: 43 Pages.
After circa 2300 years (Circle Quadrature, Archimedes, Syracuse 287 — 212 BC) the history of mathematics repeats itself in a different problem. This paper presents an original research on the Syracuse Conjecture. It proposes a theoretical development aimed at solving the conjecture through a systematic approach. One of its features suggests a process that leads to Theorem 2n+1, whose demonstration subdivides the odd numbers into seven subsets which have different behaviors applying algorithm of Collatz. It allows to replace the Collatz’s cycles with the cycles of links, transforming oscillating sequences into monotone decreasing sequences. Theorem of Independence allows to manage the cycles of links to our liking to reach very high main horizons and, when we decide, go back to lower horizons. In this article it’s proved that Collatz Conjecture is not fully demonstrable. In fact, if we consider the banal link n < 2n, there are eight cycles which connect each other in an endless of possible sequences and related links. It is a new type of Squaring of the Circle or Algebraic Quadrature, but its statement is confirmed. In other words: BIG CRUNCH (go back to 1) is always possible, but BIG BANG (to move on) has no End.
Category: Number Theory