Twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the part “twin prime” is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research paper, my attempt is to provide a valid proof for twin prime conjecture.

A twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research paper, my attempt is to provide a valid disproof for twin prime conjecture.

In the paper it is demonstrated that a valid path to a contradiction in complex number theory exists. In the path use is made of Euler’s identity and simple trigonometry. Each step can be easily verified.

This author is not a philosopher nor historian of science, but an engineering thermodynamicist. In that regard and in addition to various philosophical “why & how” treatises and existing historical analyses, the physical and logical “what it is” reflections, as successive Key Points, where a key reasoning infers the next one, along with novel contributions and original generalizations, are presented. We need to keep in mind that in Sadi Carnot’s time (early 1800s) the steam engines were inefficient (below 5%, so the heat in and out were comparable within experimental uncertainty), the conservation of caloric flourished (might be a fortunate misconception), and many critical thermal-concepts, including the conservation of energy (The First Law) were not even established. If Clausius and Kelvin were "fathers of thermodynamics" then Sadi Carnot was the "grandfather" [Kostic, 2023 July 24], or better yet, Sadi Carnot was the "Forefather of Thermodynamics-to-become" [Kostic, 2023 October 29].

This paper presents a new proof of the infinite nature of prime numbers. Grounded in fundamental Set theory and Analysis, the proof uses the method of contradiction to demonstrate the absurdity of assuming the set of all prime numbers to be finite.

This paper offers a novel geometric and visual approach to the renowned Four Color Map Theorem and K5 non-planarity problem while unveiling their profound connection to the "kissing number" problem in 2D. We represent planar graphs through circles and tangents, simplifying complex structures and shedding light on these classic problems. Our proofs by contradiction, rooted in the kissing number concept, reveal that both the Four Color Map Theorem and K5 non-planarity are fundamentally linked, as they pivot around the concept of coloring. This study bridges the realms of geometry and graph theory, providing fresh insights and emphasizing the significance of the kissing number problem in various fields.