IDENTIFY PRIME NUMBERS WITHIN INTERVALS

REGISTRO DOI: 10.70773/revistatopicos/788159015

ABSTRACT
This article analyzes the distribution and identification of prime numbers along the arithmetic scale through a deterministic filtering method in discrete intervals. The study investigates the structural properties of elements contained in the residue classes adjacent to multiples of six (6k more or less 1), isolating prime numbers from composite elements generated by factorial products and powers of prime bases. The general objective is to validate the efficacy of a screening algorithm and measure the density of primality as the upper limit of the numerical scale expands. The methodology adopts a quantitative and exploratory approach, computationally implemented through logical routines and conditional chromatic coding in Microsoft Excel software. The sample universe ranged from basal domains to a macroscopic seven-digit interval (994,008 to 1,018,080), totaling 24,072 integers. The results reveal that although the adjacency subset of 6k corresponds to 33.333% of the total domain, the progressive infiltration of composites reduces the primality rate to 7.348787% of the global interval and to 22.046361% within the 6k more or less 1 order. It is concluded that the objectives were fully achieved, confirming the hypothesis that the density of prime numbers is inversely proportional to the advancement of the arithmetic scale. The proposed model offers a low computational cost practical contribution to numerical matrix auditing and visualization in Number Theory.
Keywords: Prime Numbers; Modular Arithmetic; Electronic Spreadsheet; Algorithmic Filtering; Numerical Density.

RESUMO
Este artigo analisa a distribuição e a identificação de números primos na escala aritmética por meio de um método determinístico de filtragem em intervalos discretos. O estudo investiga as propriedades estruturais de elementos contidos nas classes de resíduos adjacentes aos múltiplos de seis (6k mais ou menos 1), isolando os números primos dos elementos compostos gerados por produtos fatoriais e potências de bases primas. O objetivo geral consiste em validar a eficácia de um algoritmo de triagem e mensurar a densidade de primariedade à medida que se expande o limite superior da escala numérica. A metodologia adotada possui abordagem quantitativa e exploratória, implementada computacionalmente por meio de rotinas lógicas e codificação cromática condicional no software Microsoft Excel. O universo amostral abrangeu desde domínios basais até um intervalo macroscópico de sete dígitos (994.008 a 1.018.080), totalizando 24.072 inteiros. Os resultados revelam que, embora o subconjunto de adjacência de 6k corresponda a 33,333% do domínio total, a infiltração progressiva de compostos reduz a taxa de primariedade a 7,348787% do intervalo global e a 22,046361% dentro das ordens 6k mais ou menos 1. Conclui-se que os objetivos foram plenamente atingidos, confirmando a hipótese de que a densidade de números primos é inversamente proporcional ao avanço da escala aritmética. O modelo proposto oferece uma contribuição prática de baixo custo computacional para a auditoria de matrizes numéricas e visualização na Teoria dos Números.
Palavras-chave: Números Primos; Aritmética Modular; Planilha Eletrônica; Filtragem Algorítmica; Densidade Numérica.

1. INTRODUCTION

The study of prime numbers and their distribution properties represents one of the most traditional and challenging theoretical frontiers in Number Theory. Historically, the search for regular patterns or deterministic mechanisms capable of rapidly isolating and classifying these elements has engaged mathematicians and computer scientists alike (Santos, 2021). Although prime numbers appear randomly scattered along the number line, an analysis of their modular properties reveals well-defined structural constraints. A notable property among these is the established fact that every prime number greater than three is necessarily confined to the residue families adjacent to multiples of six—that is, structured according to the linear forms $6k - 1$ and $6k + 1$ (Silva & Oliveira, 2020).

However, a major uncertainty and concern regarding the practical application of this property is that these adjacency classes do not contain exclusively prime elements. As the upper limit of the arithmetic scale expands, the $6k \pm 1$ coordinates become progressively "infiltrated" by composite numbers generated through the multiplication and exponentiation of lower-order prime factors—such as products of the form $(6\lambda - 1)(6\lambda + 1)$ (Garcia, 2024). This continuous infiltration creates increasing complexity in data processing and primality screening, necessitating deterministic methods that are simultaneously secure, accurate, and computationally efficient (Almeida & Costa, 2022).

In this context, the problem statement driving this research arises from the researcher's direct experience in handling and analyzing large numerical matrices within a spreadsheet environment (Microsoft Excel). Upon observing the behavior of integers across scales ranging from baseline values ​​to the order of millions (10⁶), the following research question emerged: How can a deterministic, visual method of algorithmic filtering be structured—based on excluding composite numbers in the 6k ± 1 categories—that allows for the precise identification of prime numbers and an analytical demonstration of the decrease in their density as one progresses along the numerical scale?

The rationale for this research is grounded in both its theoretical relevance and its practical applicability. From a theoretical standpoint, the study is justified by the need to deepen the understanding of prime number density dynamics within discrete partitions; it offers a clear, geometric visualization of how advancing along the arithmetic scale generates an increase in products and powers of prime bases, thereby reducing the percentage occurrence of primes within the 6k ± 1 classes.

From a practical and applied perspective, the researcher's motivation stems from the pursuit of accessible, low-computational-cost solutions for numerical data auditing. While primality testing on large datasets typically requires complex algorithms and high processing power, systematizing the mapping via conditional color-coding reduces the scope of analysis to just one-third (3.333) of the total domain (Vieira, 2021). Thus, this research demonstrates how everyday spreadsheet tools can be optimized for precise mathematical simulations, supporting both research in cryptography and number theory and teaching methodologies in the exact sciences.

To address the problem at hand and consolidate the analytical approach—moving from the general properties of modular arithmetic to a specific analysis of the seven-digit interval—this study established the following objectives:

General Objective: To propose, apply, and validate a deterministic filtering method using spreadsheets to identify prime numbers and measure their percentage density across different discrete partitions of the numerical scale.

Specific Objectives:

• Map the geometric and structural constraints regarding the occurrence of primes and twin primes within the first and second hundreds, based on the terminal digits of multiples of six;

• Implement logical routines and conditional color-coding in Microsoft Excel to isolate composite numbers from the forms (6λ ± 1) within the specific range of 994,008 to 1,018,080;

• Quantify and compare primality percentage rates across the entire range and within the 6k-adjacency subset, thereby confirming the inverse proportional relationship between prime number density and the progression of the arithmetic scale.

2. THEORETICAL FRAMEWORK OR LITERATURE REVIEW

2.1. Theoretical Framework And State Of The Art

The mathematical expression used in this study is duly registered and recorded with the National Library’s Copyright Office (Rio de Janeiro), certifying its originality and substantiating its use as a starting point for scientific inquiry. This formulation serves as the central analytical tool for defining sample intervals, enabling the identification and mapping of prime numbers with high methodological rigor and mathematical precision. Applying this model provides a structured approach to examining the arithmetic properties of the established numerical subsets.

Figure 01 – The method used to identify prime numbers with certainty and precision.

The subsequent analysis of the defined numerical intervals aims to demonstrate the model's algorithmic feasibility and analytical efficiency in screening for prime numbers. The construction of these interval partitions, based on restrictive algebraic criteria, allows the number line to be segmented into finite subsets where the distribution and density of prime numbers are systematically investigated. From a methodological standpoint, parameterizing the domain into discrete intervals enables a structured approach, mitigating the computational effort required for primality validation and increasing the degree of logical determinism within each range. Thus, the empirical validation of the data seeks to establish the statistical consistency of the proposed method, as well as its relevance to mapping the prime-counting function across the broader numerical scale.

Table 2 – identifies prime numbers in the interval from one to six.

Source: the author

The number 2 is defined as a prime number because it is the only even number with exactly two distinct positive divisors: 1 and itself. The number 3 qualifies as prime because it is the smallest odd positive integer possessing this same property, serving as the initial generator for multiples of three. Similarly, the number 5 is also classified as prime, being the first odd prime integer preceding the number 6, which represents the smallest composite multiple of three.

Within the intervals under analysis, the distribution shows that approximately 50% of the elements are prime numbers, while the remaining 50% of the terms in the aforementioned partition are composite numbers.

In the set of natural numbers, the elements 9, 15, 21, 27, 33, and 39 belong to the class of multiples of three and are, therefore, composite numbers. Conversely, the elements 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47... are formally classified as prime numbers, given that they possess exactly two distinct positive divisors in their arithmetic structure. Additionally, the number 25 is classified as a composite number because it is the square of a prime number, expressed as 5² = 25. Similarly, the number 35 is also composite, obtained through the product of two distinct prime factors via the arithmetic operation 5 × 7 = 35. Considering the interval boundary starting at 24—algebraically defined as 5² − 1 = 24—it is evident that the numbers immediately preceding or following multiples of six can be analytically represented by the linear form 6k ± 1, where k belongs to N*.

Table 3 shows multiples of three, as well as the numbers preceding and following multiples of six.

Source: the author

However, although this formation rule encompasses all prime numbers greater than or equal to five—represented in the initial sequence by the elements 7, 11, 13, 17, 19, and 23—not all integers satisfying the structural property of adjacency to multiples of six are necessarily prime. This analytical limitation is evidenced by the behavior of the number 25, which, despite strictly fitting the aforementioned linear form, is a composite number.

Thus, it is concluded that the property of preceding or succeeding multiples of six establishes a necessary but insufficient criterion for determining primality, making the application of additional divisibility tests indispensable for the definitive validation of prime elements within the domain under consideration.

In the interval under analysis, it is observed that elements adjacent to multiples of six can be algebraically represented by the linear forms (6k + 1), characterizing the successor, and (6k - 1), defining the predecessor. Such configurations encompass all candidates for primality within the domain of integers p ≥ 5. However, it must be emphasized that fitting this structural property does not, in isolation, guarantee an element's primality, making the application of complementary divisibility tests imperative for its confirmed validation.

Regarding the odd composite numbers generated by this formation rule within the interval, their occurrence stems from the multiplicative combination of odd prime factors belonging to the very residue classes adjacent to multiples of six. The analytical decomposition of these composite elements highlights the significance of minimal determining factors within the arithmetic grid, enabling the mapping of systematic distribution patterns. Consequently, the correlated examination of these numbers in relation to multiples of six and their factorial bases serves as an effective—albeit non-definitive—methodological tool for the screening and investigation of primality within specific numerical partitions. Examples:

Table 4 – shows the product of the number three and odd numbers

Source: the author

Presented below is an examination of the interval between 24 and 48, focusing on the analysis of elements immediately preceding or following the multiples of six found within that range. This procedure aims to highlight the structural property of numbers of the form 6k ± 1 within the domain under consideration—numbers that are candidates for primality.

This approach seeks to systematically examine the distribution of these numbers and their intrinsic arithmetic properties, thereby contributing to the identification and understanding of the behavior of prime numbers within this numerical set.

Table 5 – In the interval (24, 48), 25 and 35 are composite numbers.

Source: the author

In the interval between 24 and 48, a specific statistical distribution of prime and composite elements is observed, particularly within the subset of numbers adjacent to multiples of six—that is, those generated by the form 6k ± 1. Within this sample set, approximately 12.5% ​​of the numbers are composite elements defined by powers of prime numbers, as exemplified by 5² = 25. Another 12.5% ​​consist of composite numbers resulting from the product of two distinct prime factors, such as 35 = 5 × 7. Finally, 75% of the numbers immediately preceding or following multiples of six—namely 29, 31, 37, 41, 43, and 47—are formally validated as prime numbers within this interval.

This distribution reveals a geometric regularity in the occurrence of prime numbers among the predecessors and successors of multiples of six within the analyzed interval. However, it should be noted that while the form 6k ± 1 is an intrinsic property of prime numbers greater than or equal to five, it does not, in isolation, constitute a sufficient criterion to guarantee primality; complementary verification via deterministic divisibility tests is mandatory.

Presented below is an analytical examination of the predecessors and successors of multiples of six in the interval between 48 and 120. This mapping aims to examine the distribution of numbers of the form 6k ± 1 in this region of the numerical domain—numbers that possess the potential for primality.

This investigation seeks to identify patterns in the occurrence of prime and composite terms and to understand the arithmetic properties associated with these numbers. Such an approach contributes to the systematization of the proposed method and to a deeper analysis of the distribution function of prime numbers across specific intervals and the global numerical scale.

Table 6 – In this interval, multiples of 5 and 7 are composite numbers.

Source – the author

In the range between 48 and 120, the following elements are identified as prime numbers: 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, and 113. These values ​​represent approximately 20.83% of the total integers contained within the considered interval.

Regarding the composite numbers in this range, it is observed that a small fraction consists of squares of prime numbers—such as 7² = 49—representing about 1.39% of the total sample set. Additionally, composite numbers formed by the product of two distinct odd prime factors are identified—such as 55, 65, 77, 85, 91, 95, 115, and 119—accounting for approximately 9.72% of the elements in the interval.

In the context of numbers adjacent to multiples of six—that is, those structured according to the linear form 6k ± 1—it is found that the composite elements within this subset are primarily determined by the product of prime factors strictly smaller than √121 = 11. This behavior reinforces the direct influence of minimal prime factors on the composition and filtering of non-prime elements belonging to this residue class.

Continuing the distributional analysis, the interval between 120 and 168 is considered next. In this segment, particular attention is given to the terms immediately preceding or following multiples of six; these are mapped and organized into the structural subsets presented below.

Table 7 – Identification of prime numbers in the interval (120 to 168).

Source – the author

In this interval, the validated prime elements are 127, 131, 137, 139, 149, 151, 157, 163, and 167; these represent a density of approximately 56.25% of the terms structured according to the linear form 6k ± 1.

Among the composite numbers generated within this same residue class, notable occurrences include powers of prime elements—such as 121 (11²) and 125 (5³)—which account for about 6.25% of the analyzed subset. Additionally, composite numbers resulting from the product of distinct prime factors are observed—examples being 133, 143, 145, 155, and 161—totaling approximately 31.25% of the elements in the partition.

Considering the set of all integers within the 120–168 range, approximately 19.15% are prime numbers, while 80.85% are composite structures. This behavior reflects the relative asymptotic decrease in prime number density as the numerical domain expands.

Finally, an extension of the analysis to the 168–288 interval is proposed to further map the distribution of prime and composite numbers and to validate the statistical consistency of the structural patterns observed in the preceding sections.

Table 8 – This table shows the prime and composite numbers in the 168–288 interval (prime numbers + composite numbers).

Source – the author

In the interval between 288 and 360, the following elements are identified as prime numbers: 173, 307, 179, 181, 191, 193, 197, 199, 211, 227, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283. These absolute values ​​constitute the canonical set of prime numbers observed in the specified numerical partition.

When specifically considering the numbers adjacent to multiples of six—that is, those generated by the linear form 6k ± 1—it is found that approximately 43.47% of these numbers are prime, whereas an average of 56.51% are composite numbers. The composite elements mapped in this subset are: 169, 175, 185, 187, 203, 205, 209, 215, 217, 221, 235, 245, 247, 253, 259, 265, 275, 287. Among these composite terms, the perfect square 17² = 289 stands out, while the remaining elements result from products of prime factors where at least one of the factors is strictly less than the prime number 19. This result implies that such structures serve as markers for primality testing within the interval, even though a significant portion of the sample still consists of composite numbers.

Study of prime numbers in the following interval.

Table 9 – Study of numbers in the interval between 288 and 360

Source: the author

In the range between 288 and 360, the following elements are identified as prime numbers: 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, and 359. These values ​​constitute the set of prime numbers observed in the specified numerical range.

When specifically considering the numbers adjacent to multiples of six—that is, those generated by the linear form 6k ± 1—it is found that approximately 43.47% of these numbers are prime, whereas an average of 56.51% are composite numbers. The composite elements identified in this subset are: 289, 299, 301, 305, 319, 323, 325, 329, 335, 341, 343, and 355. Among these composite terms, the perfect square 17² = 289 stands out, while the remaining elements result from products of prime factors where at least one of the factors is strictly less than the prime number 19. This result suggests that such structures serve as benchmarks for primality testing within the interval, even though a significant portion of the sample still consists of composite numbers. Prime numbers in the interval under study.

Table 10 - Interval between 360 and 528

Source: the author

In the range from 360 to 528, the validated prime elements correspond to the following values: 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, and 523. Among the numbers structured in the linear form 6k ± 1 (adjacent to multiples of six), an average of 46.428571...% are prime numbers, while 53.571429...% are composite elements. Considering the totality of integers within this partition, it is observed that 15.4761190...% are prime numbers, whereas the remaining 84.523881...% are composite numbers.

The subsequent stage of the investigation involves analyzing the range from 528 to 840, covering a total span of 312 integers. As the upper limit of the numerical scale expands, an asymptotic trend of percentage decline in prime numbers—and a consequent increase in composite elements—is observed; this behavior stems from the accumulation of fundamental prime factors, which raises the density of composite numbers within the 6k ± 1 residue classes, thereby increasing the occurrence of non-prime values ​​within the sample domain. The prime numbers identified in this interval (528 to 840) are mapped to the subset: {541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839}. This group totals 46 prime elements, equivalent to 14.743589...% of all integers in the range, leaving a fraction of 85.256411...% as composite numbers for the overall interval. Restricting the analytical scope strictly to the terms preceding or following multiples of six within this partition, the distribution shows 42.592592...% prime elements versus 57.4074...% composite structures.

We will now demonstrate the prime numbers in the next interval, (840 to 960).

Table 11 – Interval between 528 and 840

Source – the author

We will demonstrate the prime numbers in the next interval (840 to 960).

Table 12 – The interval between 840 and 960

Source – the author

The prime elements identified in the interval under analysis correspond to the following values: 853, 857, 859, 863, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, and 953. Restricting the analytical scope exclusively to the numbers adjacent to multiples of six (of the form 6k ± 1), it is found that 40% are prime numbers, whereas the remaining 60% constitute composite structures. From the perspective of the overall domain of the partition in question, prime numbers represent 13.33333...% of the total terms, while composite elements account for 86.666667...%.

In the next interval shown, it is observed that the composite elements situated between the predecessors and successors of multiples of six are multiples of the prime numbers within the range extending from the number 5—the initial element of the set of integers preceding a multiple of 6—to the prime number 31, which marks the lower limit of the section under study. The prime numbers mapped in the following spreadsheet are: 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, and 1367. In this sample region, there are 57 prime elements. This volume corresponds to 13.97058823...% of the total integers in the overall interval. When isolating only the terms generated by the residue classes 6k ± 1, the distribution shows 41.91176470...% prime numbers versus 58.0882353...% composite structures.

Let us move to the interval (1368 – 1680). Tables 13 and 14 – interval between 960 and 1368

Let's move to the interval (1368–1680).

Source – the author

In the interval under analysis, the following 44 prime elements are identified: 1373, 1381, 1399, 1409, 1423, (1427, 1429), 1433, 1439, 1447, (1451, 1453), 1459, 1471, (1481, 1483), (1487, 1489), 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, (1607, 1609), 1613, (1619, 1621), 1627, 1637, 1657, 1663, and (1667, 1669). Restricting the analytical scope strictly to the numbers generated by the residue classes adjacent to multiples of six (of the linear form 6k ± 1), this sample quantifies a density of approximately 42.307692...% prime numbers versus 52.692308...% composite structures.

Table 15 – interval between 1680 and 1848

Source: the author

In the interval spanning 1680 to 1848, the following prime numbers are identified: 1693, (1697, 1699), 1709, (1721, 1723), 1733, 1741, 1747, 1753, 1759, 1777, 1783, (1787, 1789), 1801, 1811, 1823, 1831, and 1847. When the analytical scope is restricted solely to terms generated by residue classes adjacent to multiples of six (of the linear form 6k ± 1), a distribution of 35.714285...% prime numbers versus 64.285715...% composite structures is observed. From a macroscopic perspective of the entire numerical domain contained within this partition, prime numbers represent 11.904761...% of the total, while the remaining 88.095239...% consist of composite numbers.

Table 16 – Interval between 1848 and 2208

Source – the author

In the adopted analytical matrix (spreadsheet) representation model, a distinct color-coding scheme is applied to segregate the elements: prime numbers are highlighted in yellow, while composite numbers are assigned different colors based on their factorial properties. This visual differentiation aims to map the structural specificities of the composite numbers—whether they are products of distinct prime factors or powers—which emerge exclusively within the residue classes adjacent to multiples of six (following the linear form 6k ± 1). Consequently, a multi-factorial, multi-phase system is established for the rapid identification of prime and composite elements within this distribution. Additionally, it should be noted that even numbers and multiples of three were excluded from the spreadsheet's sampling scope, as these classes do not belong to the residue families relevant to mapping the primality and structural composition under analysis. Set of prime numbers: {2,(3, 5), (5, 7), (11, 13), (17, 19), 23, (29, 31), 37, (41, 43), 47, 53, (59, 61), 67, (71, 73), 79, 83, 89, 97, (101, 103), (107, 109), 113, 127, 131, (137, 139), (149, 151), 157, 163, 167, 173, (179, 181), (191, 193), (197, 199) 241), 251, 257, 263, (269, 271), 277, (281, 283), 293, 307, (311, 313), 317, 331, 337, (347, 349) 383, 389, 397, 401, 409, (419, 421), 431, 439, 443, 449, 457, (461, 463), 467, 479, 487, 491, 499, 503, 509, (521, 523), 541, 547, 557, 563, 569, 571, 577, 587, 593, (599, 601), 607, 613, (617, 619), 631, (641, 643), 647, 653, (659, 661), 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, (809, 811), (821, 823), (827, 829), 839, 853, (857, 859), 863, (881, 883), 887, (907, 911), 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, (1031, 1033), 1039, (1049, 1051), (1061, 1063), 1069, 1087, (1091, 1093), 1097, 1103, 1109, 1117, 1123, 1129, (1151, 1153), 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, (1229, 1231), 1237, 1249, 1259, (1277, 1279), 1283, (1289, 1291), 1297, (1301, 1303), 1307, (1319, 1321), 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, (1427, 1429), 1433, 1439, 1447, (1451, 1453), 1459, 1471, (1481, 1483), (1487, 1489), 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, (1607, 1609), 1613, (1619, 1621), 1627, 1637, 1657, 1663, (1667, 1669), 1693, (1697, 1699), 1709, (1721, 1723), 1733, 1741, 1747, 1753, 1759, 1777, 1783, (1787, 1789), 1801, 1811, 1823, 1831, 1847, 1861, 1867, (1871, 1873), (1877, 1879), 1889, 1901, 1907, 1913, (1831, 1933), (1949, 1951), 1973, 1979, 1987, 1993, (1997, 1999), 2003, 2011, 2017, (2027, 2029), 2039, 2053, 2063, 2069, (2081, 2083), (2087, 2089), 2099, (2111, 2113), (2129, 2131), 2137, (2141, 2143), 2153, 2161, 2179, 2203, 2207,

This paper presents a deterministic method for identifying prime numbers, characterized by high accuracy and analytical consistency. The applicability of the proposed algorithm was empirically validated up to the order of magnitude of 10⁶ (the millions range), demonstrating potential scalability for higher numerical domains. The system's computational robustness ensures the integrity of the results obtained, mitigating processing biases. Any inconsistencies in the classification of elements within specific intervals do not stem from limitations intrinsic to the theoretical model, but rather from human operational variables associated with data entry or manipulation. Consequently, rigorous control of input parameters and systematic auditing of processes are mandatory steps to safeguard the precision, reproducibility, and reliability of the numerical partitions established by the method. Showing prime numbers in the hundreds.

Table 17 – Prime numbers in the first hundred.

Source – the author

The distribution of prime numbers within the first hundred integers is presented here, segmented by their constituent linear partitions: the first interval (1–6), the second (6–25), the third (25–48), and the fourth (48–120). Within this initial elementary domain, composite numbers coexist—comprising even numbers, multiples of three, and products or powers of prime bases, specifically: (25, 35, 49, 55, 65, 77, 85, 91, 95). To optimize and deterministically simplify the process of identifying these composite structures, a discrete interval analysis methodology is adopted. In the second hundred—corresponding to the domain [100, 200]—the set of primes spans the terminal section of the fourth interval (48–120), the entirety of the fifth interval (120–168), and the initial section of the sixth interval (168–288). It is noteworthy that, within the fully mapped intervals, arithmetic properties behave consistently, validating the uniform application of this same methodological approach to all elements of the partition.

In tens where multiples of six end with the digits 2 or 8 in the units place, a symmetrical pattern emerges involving two preceding and two succeeding elements adjacent to these multiples; this configuration offers the potential for up to four prime numbers, enabling the formation of two pairs of twin primes. Conversely, in tens where multiples of six have units digits of 4 or 0, the structure is limited to two preceding elements and only one succeeding element, restricting the count of primes to a maximum of two and precluding the occurrence of twin primes within that range. Similarly, for the tens groups where multiples of six have units in the range (0–6), two successor terms and only one adjacent predecessor are identified, resulting in up to two prime numbers and the absence of twin primes. This concludes the systematic demonstration of the prime elements belonging to the second hundred.

Table 18 – Prime numbers in the second hundred

Source – the author

Table 19 – range 12.768 to 16.128