Geometric Integration of Twin Primes and Sophie Germain Primes via the Period-60 of SP Numbers in a Multiple Fullerene Model

Geometric Integration of Twin Primes and Sophie Germain Primes via the Period-60 of SP Numbers in a Multiple Fullerene Model Author: Masaki Tsuchimoto [Abstract] This paper presents a novel mathematical model that geometrically integrates and approaches the unsolved problems of the infinitude of twin primes and Sophie Germain primes in number theory, utilizing the properties of SP numbers (Sum is Prime) and the geometric structure of C60 fullerenes. We analyze the conditions for SP numbers of the form $n=2p$ (where $p$ is an odd prime) and mathematically prove that they strictly exist within the congruence classes $n \equiv 22 \pmod{60}$ and $n \equiv 58 \pmod{60}$. This period of 60 is spatially isomorphic to the 60 vertices of a C60 fullerene. Furthermore, we show that the process of avoiding numerical collapse into multiples of 3 or 5 (loss of primality) is geometrically equivalent to the Isolated Pentagon Rule (IPR), which ensures fullerene stability. This paper demonstrates that the independent conjectures of twin primes and Sophie Germain primes are integrated through a single symmetry filter—the "period-60" and "multiple fullerene geometry"—revealing the mathematical foundation for the fundamental principle of information preservation in the realm of numbers. Chapter 1: Introduction and Mathematical Proof of the "Period-60" of SP Numbers 1.1 Introduction: SP Numbers as the Integration of Unsolved Problems The distribution of prime numbers often appears chaotic and disordered. Among them, the questions of whether there are infinitely many "pairs of primes differing by 2 (twin primes)" and "pairs where both $p$ and $2p+1$ are primes (Sophie Germain primes)" have perplexed mathematicians for centuries. This paper focuses on "SP numbers (Sum is Prime: numbers where the sum of any two factors is always prime)" as unique numbers that simultaneously satisfy these two independent problems. In this chapter, we focus on the ultimate SP numbers with the fewest prime factors, taking the form $n=2p$ (where $p$ is an odd prime greater than 3), and unravel their mathematical structure. 1.2 Absolute Conditions for $n=2p$ Type SP Numbers and Anomaly Avoidance The only combinations of two integers that multiply to $n=2p$ are $(1 \times 2p)$ and $(2 \times p)$. By the definition of SP numbers, the three numbers (I) $p$, (II) $p + 2$, and (III) $2p + 1$ must simultaneously be prime. As numbers grow larger, an anomaly arises where they collapse into "multiples other than primes." In modulo 3, if we assume $p \equiv 1 \pmod 3$, then $p + 2 \equiv 0 \pmod 3$, leading to a collapse. Therefore, $p \equiv 2 \pmod 3$ is strictly required. Furthermore, given the condition that $p$ is an odd number, $p \equiv 5 \pmod 6$ is established in modulo 6. Next, in modulo 5, if $p \equiv 2 \pmod 5$, then $2p + 1 \equiv 0 \pmod 5$, and if $p \equiv 3 \pmod 5$, then $p + 2 \equiv 0 \pmod 5$, both resulting in a collapse. Thus, $p$ must be $p \equiv 1$ or $4 \pmod 5$. 1.3 Integration into Period 60 and Determination of Singularities Integrating the anomaly avoidance conditions above, we determine the periodicity of $p$ using the least common multiple, modulo 30. Among the congruence classes in modulo 30 that satisfy the modulo 6 condition, namely $\{5, 11, 17, 23, 29\}$, only $11$ and $29$ satisfy the modulo 5 condition. Therefore, $p \equiv 11 \pmod{30}$ or $p \equiv 29 \pmod{30}$. Since the SP numbers we are seeking are of the form $n = 2p$, converting this to modulo 60 yields: $n \equiv 22 \pmod{60}$ or $n \equiv 58 \pmod{60}$. Thus, it is mathematically proven that SP numbers of the form $n=2p$ greater than 10, as a result of continuously avoiding the anomaly of prime collapse, strictly exist only in the congruence classes of '22' or '58' in modulo 60. Chapter 2: Geometric Projection onto C60 Fullerene Geometry and the Mechanism of Information Preservation 2.1 Topological Isomorphism between Modulo 60 Arithmetic and Fullerene Geometry We reinterpret the mathematically proven "period-60" as a geometric structure in three-dimensional space. In nature, the most highly symmetric structure composed of 60 units is the "C60 fullerene". We define a projection operator $\hat{F}_{60} | n \rangle = | n \bmod 60 \rangle$ that maps the straight line of integers onto the 60 vertices constituting the C60 fullerene. Through this, it becomes evident that the infinitely existing candidates for SP numbers accumulate exclusively on just two singularities on the fullerene, $V_{22}$ and $V_{58}$. 2.2 Geometric Cancellation of Anomalies via the Isolated Pentagon Rule (IPR) Among the numbers projected onto $V_{22}$ and $V_{58}$, the process in which primality collapses due to higher-order factors (such as multiples of 7 or 11) is geometrically equivalent to the "Isolated Pentagon Rule (IPR)" that governs the stability of C60 fullerenes. In IPR, the adjacency of pentagons means the collapse of the structure. The emergence of new prime factors in the realm of numbers corresponds to this "pentagonal distortion." Discovering an SP number is nothing less than identifying "an absolutely stable coordinate completely surrounded by hexagons, without sharing an edge with any anomaly (pentagon)" within a fractally expanding polyhedral geometry. 2.3 Multiple Fullerene Cosmological Model and the Preservation of Prime Information (Conclusion) This geometric approach implies that numbers satisfying the SP condition $\Psi_{SP}(n)$ continue to be projected onto the central axis of a dimensionally expanded multiple fullerene (carbon onion) structure. Under conventional methods of blindly searching along the number line, the appearance of twin primes and Sophie Germain primes seems to be a probabilistic chaos. However, the moment it is projected onto the multiple fullerene structure, it transforms into a deterministic geometric process of "anomaly cancellation and symmetry acquisition." Contradictory factors are completely eliminated by an IPR-like rule, and only pure prime information is preserved into the next dimension through the singularities. This study demonstrates that the independent unsolved problems of twin primes and Sophie Germain primes are integrated by passing through the geometric symmetry filter of C60, thereby proving the absolute mechanism of information preservation in the world of numbers.