The 2n+3 Sieve: a Congruence-Based Representation of Odd Divisibility using Collision Matrices

The 2n+3 Sieve: a Congruence-Based Representation of Odd Divisibility using Collision Matrices

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  THE 2n +3 SIEVE: A CONGRUENCE-BASED REPRESENTATION OF ODD DIVISIBILITY USING COLLISION MATRICES GIANLUCA REMIGIO PISANO Abstract. We introduce a congruence-based representation of odd pri- mality built upon the affine transformation f(n) = 2n + 3, which establishes a bijection between the set of non-negative integers and the odd integers greater than or equal to three. Instead of analysing odd integers directly, divisibility is transferred to the index space, where composite numbers are characterized by simple modular congruences. This formulation naturally induces an infinite binary structure, called the Collision Matrix, whose rows correspond to odd divisors and whose columns represent odd integers. We prove that the absence of collisions below the classical square-root threshold is equivalent to primality. Structural properties of the matrix, including periodicity, arithmetic progressions, and congruence classes, are investigated. Finally, we discuss algorithmic implications and compare this repre- sentation with classical deterministic sieve constructions. 1. Introduction The distribution of prime numbers remains one of the central topics of elementary and computational number theory. Since every prime greater than two is odd, many primality algorithms restrict their search space to odd integers. The present work follows this philosophy but adopts a different viewpoint. Instead of considering odd integers directly, we introduce a dense index space N0 = {0, 1, 2, . . .}, together with the affine transformation Date: August 4, 2026. 2020 Mathematics Subject Classification. 11A41, 11Y11, 68Q25 . Key words and phrases. Prime numbers, primality testing, collision matrix, modular arithmetic, number theory, sieve algorithms . 1
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  2 GIANLUCA REMIGIOPISANO f(n) = 2n + 3. This mapping establishes a one-to-one correspondence between indices and odd integers. The principal observation is that divisibility in the odd integers translates into simple congruence relations within the index space. Consequently, composite numbers appear as periodic collision patterns, which can be organized into a binary infinite matrix. This matrix, called the Collision Matrix, constitutes the main mathemat- ical object studied throughout the paper. Unlike classical sieve algorithms, our emphasis is placed on the algebraic representation of divisibility rather than on implementation details. The resulting framework provides a unified description of congruence classes, periodic structures, and primality testing. 2. Preliminaries Throughout the paper, N0 = {0, 1, 2, . . .}, and O = {3, 5, 7, 9, . . .} denotes the set of odd integers greater than or equal to three. 2.1. Affine Index Representation. Definition 2.1. Define f : N0 −→ O by f(n) = 2n + 3. Proposition 2.2. The mapping f(n) = 2n + 3 is a bijection. Proof. Injectivity follows immediately from 2n1 + 3 = 2n2 + 3 =⇒ n1 = n2. Surjectivity is immediate, since every odd integer greater than or equal to three can be uniquely written as 2n + 3
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  THE 2N+3 SIEVE3 for some n ∈ N0. □ 2.2. Index Representation of Divisibility. The central idea is that di- visibility admits a remarkably simple description within the index space. Rather than testing 2n + 3 directly, we characterize divisibility through congruence classes on the indices. This observation constitutes the foundation of the collision framework developed in the following sections. 3. The Collision Matrix The affine representation introduced in the previous section transfers the divisibility structure of odd integers into the index space. Each odd divisor generates a periodic family of indices corresponding ex- actly to its odd multiples. This observation motivates the introduction of a binary matrix whose entries encode divisibility relations. 3.1. Collision Sets. Definition 3.1. Let di = 2i + 3, i ∈ N0. The collision set generated by the divisor di is Ci = {k ∈ N0 : k ≡ i (mod 2i + 3)} . Every element of Ci corresponds to an odd integer divisible by di. Indeed, k = i + m(2i + 3), implies 2k + 3 = (2m + 1)(2i + 3). Therefore, Ci = {i, i + (2i + 3), i + 2(2i + 3), . . .}, which is an arithmetic progression with common difference 2i + 3.
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  4 GIANLUCA REMIGIOPISANO 3.2. The Collision Matrix. Definition 3.2. The Collision Matrix M = (mij)i,j≥0 is defined by mij =    1, j ≡ i (mod 2i + 3), 0, otherwise. The i-th row represents the action of the odd divisor 2i + 3, while the j-th column corresponds to the odd integer 2j + 3. Hence, mij = 1 if and only if 2i + 3 | 2j + 3. Figure 1. Binary structure of the Collision Matrix M. Rows correspond to odd divisor operators di = 2i + 3, and columns represent index values j ∈ N0. The highlighted column n = 4 corresponds to N = 2(4) + 3 = 11. Below the square-root bound (2i + 3 ≤ √ 11 ≈ 3.31, corresponding to row i = 0), no collision occurs (m0,4 = 0), illustrating the prime columns property.
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  THE 2N+3 SIEVE5 Figure 1 illustrates the binary structure of the Collision Matrix for the initial divisor rows and column indices. As observed in the highlighted col- umn (n = 4), the unique collision below the diagonal occurs at row i = 4 (d4 = 11), representing the trivial self-divisibility 11 | 11. Below the classical square-root threshold ( √ 11 ≈ 3.31), the only relevant row is i = 0 (d0 = 3). Since m0,4 = 0, the column is free of collisions within the search bound, confirming that N = 11 is prime according to Theorem 4.4. Figure 2. Extended view of the Collision Matrix M for n = 15 (N = 2(15) + 3 = 33) across 26 total column indices (j ∈ {0, . . . , 25}). The highlighted column n = 15 exhibits a collision at row i = 0 (d0 = 3 ≤ √ 33 ≈ 5.74), which immediately confirms that N = 33 is composite since 3 | 33. Additional collisions at rows i = 0 and i = 4 (d4 = 11) further reflect its prime factorization 33 = 3 × 11. Figure 2 displays the Collision Matrix extended over 26 column indices (j ∈ {0, . . . , 25}). Focusing on the highlighted column n = 15, corresponding to N = 33, a collision is present at row i = 0 (d0 = 3). Since d0 = 3 ≤ √ 33 ≈ 5.74, Theorem 4.4 implies that N = 33 is composite. The additional collision at row i = 4 (d4 = 11) illustrates how multiple non-zero entries encode the full set of odd divisors of N. 3.3. Periodicity. The collision matrix possesses a remarkably regular struc- ture. Each row consists of infinitely many periodically spaced collisions. Proposition 3.3. For every fixed row i, the support supp(Mi) = {j : mij = 1} is an arithmetic progression i, i + (2i + 3), i + 2(2i + 3), . . . having minimal period Ti = 2i + 3.
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  6 GIANLUCA REMIGIOPISANO Proof. By definition, mij = 1 precisely when j = i + m(2i + 3) for some integer m ≥ 0. Successive values differ by 2i + 3, which proves periodicity. Minimality follows because no smaller positive increment preserves the congruence class. □ 3.4. Row Density. Since one collision appears every 2i + 3 columns, the asymptotic density of row i is immediately obtained. Proposition 3.4. The natural density of the collision set generated by row i equals δi = 1 2i + 3 . Proof. Among the first N columns, the number of collisions satisfies |Ci ∩ [0, N]| = N 2i + 3 + O(1). Dividing by N and letting N → ∞ gives δi = lim N→∞ |Ci ∩ [0, N]| N = 1 2i + 3 . □
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  THE 2N+3 SIEVE7 3.5. Column Interpretation. Rows describe divisibility operators. Columns describe individual odd integers. For a fixed index n, consider the column M[:, n]. Each non-zero entry corresponds to an odd divisor of 2n + 3. Consequently, the number of collisions within a column coincides with the number of odd divisors represented inside the inspected portion of the matrix. This observation naturally leads to a criterion for primality, which is es- tablished in the next section. 4. Fundamental Theorems The collision matrix provides an algebraic representation of divisibility within the affine index space. We now establish the fundamental results upon which the entire frame- work is built. 4.1. Congruence Characterization. Theorem 4.1 (Congruence Characterization). Let f(n) = 2n + 3. For every pair of indices i, n ∈ N0, the following statements are equivalent: 2i + 3 | 2n + 3, and n ≡ i (mod 2i + 3). Proof. Suppose 2i + 3 | 2n + 3. Then there exists an odd integer q = 2m + 1 such that 2n + 3 = (2i + 3)(2m + 1).
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  8 GIANLUCA REMIGIOPISANO Expanding, 2n + 3 = 4im + 6m + 2i + 3. Subtracting three and dividing by two, n = 2im + 3m + i = m(2i + 3) + i. Hence, n ≡ i (mod 2i + 3). Conversely, assume n = i + m(2i + 3). Then 2n + 3 = 2(i + m(2i + 3)) + 3 = 2m(2i + 3) + 2i + 3 = (2m + 1)(2i + 3), showing that 2i + 3 | 2n + 3. Therefore, the two conditions are equivalent. □ 4.2. Collision Criterion. The previous theorem immediately yields a ma- trix interpretation. Corollary 4.2. For every pair (i, n), one has mi,n = 1 if and only if 2i + 3 | 2n + 3.
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  THE 2N+3 SIEVE9 4.3. Square-Root Bound. The classical square-root stopping criterion ad- mits a particularly simple formulation within the collision matrix. Theorem 4.3. Suppose N = 2n + 3. If N is composite, then there exists an index i such that mi,n = 1 and 2i + 3 ≤ √ N. Proof. Assume N is composite. Elementary number theory guarantees the existence of a prime divisor p ≤ √ N. Since N is odd, its prime divisor is also odd, hence p = 2i + 3 for some i ∈ N0. Applying Theorem 4.1, mi,n = 1. This completes the proof. □
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  10 GIANLUCA REMIGIOPISANO 4.4. Primality Criterion. We now obtain the principal characterization of prime numbers. Theorem 4.4 (Primality Criterion). Let N = 2n + 3. Then N is prime if and only if mi,n = 0 for every index satisfying 2i + 3 ≤ √ N. Proof. Suppose first that N is prime. No odd divisor greater than one can divide N. Therefore, no collision occurs, and every inspected entry is zero. Conversely, suppose every inspected entry below the square-root bound is zero. If N were composite, Theorem 4.2 would guarantee the existence of a collision. This contradicts the hypothesis. Hence, N must be prime. □
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  THE 2N+3 SIEVE11 4.5. Uniqueness of Prime Columns. Prime numbers admit a remarkably simple interpretation within the collision matrix. Corollary 4.5. The column associated with N = 2n + 3 contains no collisions below the square-root threshold if and only if N is prime. 4.6. Finite Verification. Although the collision matrix is infinite, only finitely many rows must be inspected. Proposition 4.6. The number of inspected rows equals √ 2n + 3 − 3 2 + 1. Proof. The stopping condition is 2i + 3 ≤ √ 2n + 3. Solving for i gives i ≤ √ 2n + 3 − 3 2 . Since i is integral, the result follows immediately. □ 5. Algebraic Structure of the Collision Matrix The Collision Matrix is more than a computational device. Its entries define a family of arithmetic progressions whose interaction induces a rich algebraic structure. In this section we investigate translation invariance, periodicity, and the interaction among collision sets.
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  12 GIANLUCA REMIGIOPISANO 5.1. Translation Invariance. Each collision row is generated by a congru- ence class. Consequently, the pattern is invariant under translations equal to its pe- riod. Theorem 5.1 (Translation Invariance). Let Ti = 2i + 3. Then mi,j = mi,j+Ti for every j ≥ i. Proof. By definition, mi,j = 1 if and only if j ≡ i (mod Ti). Since j + Ti ≡ j (mod Ti), the congruence class is preserved. Hence mi,j = mi,j+Ti . □ 5.2. Arithmetic Progression Structure. The support of every row forms an arithmetic progression. Theorem 5.2. For every divisor index i, the support supp(Mi) is { i + k(2i + 3) | k ≥ 0 }.
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  THE 2N+3 SIEVE13 Proof. Immediate from the definition j ≡ i (mod 2i + 3). □ 5.3. Intersection of Collision Sets. Different rows intersect only when their congruence systems are simultaneously satisfiable. Theorem 5.3. Let Ci = {j : j ≡ i (mod 2i + 3)} and Cr = {j : j ≡ r (mod 2r + 3)}. Then Ci ∩ Cr ̸= ∅ if and only if i − r ≡ 0 (mod gcd(2i + 3, 2r + 3)). Proof. This is the classical compatibility condition for two simultaneous con- gruences. The result follows directly from the Chinese Remainder Theorem. □ 5.4. Chinese Remainder Decomposition. Whenever two collision rows intersect, their common collisions form another arithmetic progression. Proposition 5.4. Suppose Ci ∩ Cr ̸= ∅. Then Ci ∩ Cr = {a + kL}, where L = lcm(2i + 3, 2r + 3). Proof. The Chinese Remainder Theorem guarantees a unique residue class modulo L. All common collisions belong to this class. □
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  14 GIANLUCA REMIGIOPISANO 5.5. Global Periodicity. Although the Collision Matrix is infinite, every finite collection of rows admits a global period. Theorem 5.5. Let M(r) denote the restriction of the Collision Matrix to its first r rows. Then M(r) is periodic with global period Pr = lcm(3, 5, 7, . . . , 2r + 3). Proof. Each row possesses period 2i + 3. The least common multiple of all row periods preserves every row simul- taneously. Therefore, Pr = lcm(3, 5, . . . , 2r + 3). □ 5.6. Prime Columns. Prime numbers admit a geometric interpretation in- side the Collision Matrix. Definition 5.6. Let n ∈ N0. The column M[:, n] is called a prime column whenever mi,n = 0 for every row satisfying 2i + 3 ≤ √ 2n + 3.
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  THE 2N+3 SIEVE15 5.7. Characterization of Prime Columns. Theorem 5.7. The integer 2n + 3 is prime if and only if its corresponding column is a prime column. Proof. Immediate from the Primality Criterion established in Section 4. □ 5.8. Sparse Structure. The Collision Matrix is asymptotically sparse. Proposition 5.8. The density of ones in row i equals 1 2i + 3 . Hence, the average density of the first r rows satisfies 1 r r−1 X i=0 1 2i + 3 = O log r r . Consequently, the Collision Matrix becomes asymptotically sparse. Proof. The first statement was proved previously. The second follows from the asymptotic behaviour of the harmonic series restricted to odd integers. □ Proposition 5.9 (Implicit Collision Matrix). The explicit construction of the Collision Matrix is not required for the deterministic primality test. Indeed, for a given odd integer N = 2n + 3, the primality criterion depends exclusively on the evaluation of the con- gruence conditions n ≡ i (mod 2i + 3), for all admissible divisor indices satisfying
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  16 GIANLUCA REMIGIOPISANO 2i + 3 ≤ √ N. The Collision Matrix provides a structured representation of these congru- ence classes, but the algorithm operates directly on the corresponding residue relations without constructing the matrix explicitly. Consequently, the matrix should be regarded as a mathematical model of odd divisibility rather than a computational prerequisite for the algorithm. Proof. Each row of the Collision Matrix represents exactly one residue class n ≡ i (mod 2i + 3). The primality algorithm evaluates these congruence relations individually until either a collision is detected or all admissible divisors up to the square- root bound have been examined. Since every matrix entry is generated by the same congruence relation, constructing the matrix explicitly provides no additional arithmetic infor- mation beyond the residue tests already performed by the algorithm. Therefore, the explicit matrix is unnecessary for correctness and serves only as a conceptual and visual representation of the underlying divisibility structure. The preceding results show that the Collision Matrix is naturally endowed with an algebraic structure inherited from congruence classes and their arith- metic progressions. The matrix should therefore be viewed as a discrete algebraic object rather than merely as a computational table. □ 6. Examples The following examples illustrate how the Collision Matrix encodes odd divisibility through congruence classes. 6.1. Example 1: A Prime Number. Consider N = 37. Since 37 = 2 · 17 + 3, its corresponding column is n = 17. The square-root bound is √ 37 ≈ 6.08, therefore only the divisor indices
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  THE 2N+3 SIEVE17 i = 0 (d = 3) and i = 1 (d = 5) must be inspected. The corresponding congruences are 17 ̸≡ 0 (mod 3), and 17 ̸≡ 1 (mod 5). No collision occurs. Hence, 37 is correctly classified as prime. 6.2. Example 2: A Composite Number. Consider N = 35. Since 35 = 2 · 16 + 3, its index is n = 16. Again, √ 35 ≈ 5.91, so only the rows d = 3 and d = 5 must be examined. The congruence 16 ≡ 1 (mod 5) holds, which corresponds to the collision
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  18 GIANLUCA REMIGIOPISANO m1,16 = 1. Therefore, 5 | 35, and 35 is composite. 6.3. Visual Interpretation. Figure 1 e 2 illustrates a finite portion of the Collision Matrix. Rows correspond to odd divisors, whereas columns correspond to odd integers. Every highlighted entry represents a collision, that is, 2i + 3 | 2n + 3. Prime numbers appear as columns containing no collisions below the square-root threshold. The highlighted column associated with n = 15 illustrates the divisibility pattern of the corresponding odd integer within the matrix. 7. Algorithm The primality test follows directly from the Collision Criterion. Given an index n, corresponding to the odd integer N = 2n + 3, only divisor indices satisfying 2i + 3 ≤ √ N must be inspected.
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  THE 2N+3 SIEVE19 Algorithm 1 The 2n + 3 Primality Test Require: n ≥ 0 Ensure: TRUE iff 2n + 3 is prime 1: for i = 0 to ( √ 2n + 3 − 3)/2 do 2: if n ≡ i (mod 2i + 3) then 3: return FALSE 4: end if 5: end for 6: return TRUE 8. Correctness Theorem 8.1. Algorithm 1 returns TRUE if and only if 2n + 3 is prime. Proof. The result follows directly from the Primality Criterion established in Theorem 4.4. Every composite odd integer possesses an odd divisor not exceeding its square root. Conversely, if no collision occurs below this bound, no non-trivial odd divisor exists, and the integer is prime. □ 9. Complexity For a single odd integer N = 2n + 3, the number of inspected rows equals $√ N − 3 2 % + 1, yielding O( √ N) modular congruence tests. The algorithm requires constant auxiliary memory, O(1), since only the current divisor index is stored. The Collision Matrix itself is conceptual and need not be materialized explicitly.
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  20 GIANLUCA REMIGIOPISANO 10. Computational Performance The implementation distinguishes between two computational tasks: (1) primality verification of a single odd integer through the Collision Matrix criterion; (2) construction of a finite Collision Matrix for visualization purposes. These tasks have different computational objectives and therefore different complexity characteristics. 10.1. Primality Verification. Given an odd integer N = 2n + 3, the algorithm inspects only divisor indices satisfying 2i + 3 ≤ √ N. For each admissible divisor, a single congruence n ≡ i (mod 2i + 3) is evaluated. As soon as the first collision is detected, the computation terminates im- mediately. Consequently, the number of modular evaluations is bounded by O( √ N), while composite integers frequently require considerably fewer iterations because of the early stopping criterion. 10.2. Construction of the Collision Matrix. The finite Collision Matrix is generated independently of the primality test. Each row is produced through arithmetic progressions of constant step 2i + 3, marking all indices belonging to the corresponding residue class. The matrix is intended as a structural representation of odd divisibility rather than as an optimized sieve data structure. 10.3. Implementation Remarks. The reference implementation records two execution times. The first measures only the deterministic primality test. The second measures the construction of the visualization matrix. These measurements are reported separately because matrix generation is not required for primality testing itself. Execution times depend on the hardware platform, the Python interpreter, and compiler optimizations.
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  THE 2N+3 SIEVE21 Therefore, they should be regarded as implementation benchmarks rather than intrinsic properties of the proposed mathematical framework. The principal contribution of the present work is the congruence-based representation of odd divisibility, whereas the measured execution times merely illustrate the practical feasibility of the algorithm. 11. Experimental Results Experimental measurements reported in the accompanying implementa- tion indicate that the congruence formulation provides an efficient determin- istic realization of odd primality testing over the numerical ranges investi- gated. The benchmarks presented in this work should be interpreted as implementation- dependent measurements rather than asymptotic comparisons with other primality algorithms. 12. Relation to Classical Sieve Theory The framework developed in this paper should be interpreted as an al- ternative representation of odd divisibility rather than as a replacement for classical sieve algorithms. Classical sieves, such as the Sieve of Eratosthenes, operate by progressively eliminating multiples of previously identified prime numbers from a finite interval. Their computational objective is the enumeration of primes. The present construction adopts a different point of view. Instead of removing multiples, divisibility is represented explicitly through congruence classes defined in the affine index space f(n) = 2n + 3. Each odd divisor generates a periodic collision set, Ci = {n : n ≡ i (mod 2i + 3)}, and the collection of all collision sets forms the Collision Matrix. Within this representation, • rows correspond to odd divisors; • columns correspond to odd integers; • collisions correspond exactly to divisibility relations; • primality is characterized by the absence of collisions below the square- root bound. Consequently, the Collision Matrix provides an explicit algebraic model of odd divisibility, whereas classical sieve algorithms describe an elimination process. The two viewpoints are therefore complementary. One emphasizes algorithmic filtering, whereas the other emphasizes the underlying congruence structure.
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  22 GIANLUCA REMIGIOPISANO From this perspective, the principal contribution of the present work lies in the mathematical representation of divisibility, rather than in claiming an asymptotic improvement over existing deterministic primality tests. The Collision Matrix may therefore be regarded as a discrete algebraic ob- ject worthy of investigation independently of its computational applications. 13. Discussion The principal contribution of this work is not a new asymptotic complexity result, but rather a different algebraic representation of odd divisibility. The affine transformation f(n) = 2n + 3 moves primality testing from the integer domain to a congruence space in which divisibility is represented by periodic collision sets. This representation naturally induces the Collision Matrix, whose rows correspond to congruence classes generated by odd divisors. Beyond its computational interpretation, the matrix exhibits structural properties including periodicity, translation invariance, and arithmetic pro- gression supports, suggesting possible connections with other discrete math- ematical structures. 14. Geometric Interpretation Although the Collision Matrix is introduced as an algebraic representation of odd divisibility, it also admits several equivalent geometric interpretations. These visual representations do not modify the underlying arithmetic cri- terion, but provide complementary insight into the organization of the con- gruence classes generated by the affine map N = 2n + 3. 14.1. Arithmetic Progressions. For each divisor index i ≥ 0, the corresponding residue class is n ≡ i (mod 2i + 3), which generates the arithmetic progression n = i + m(2i + 3), m ≥ 0. Each progression identifies all odd integers divisible by 2i + 3. These progressions constitute the rows of the Collision Matrix.
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  THE 2N+3 SIEVE23 The congruence classes introduced in the previous sections admit multiple equivalent geometric representations. Figure 3 illustrates three complementary visualizations generated from the same family of arithmetic progressions. Although geometrically distinct, all representations encode exactly the same divisibility information. Figure 3. Equivalent geometric representations of the Col- lision Matrix. (a) Collision Matrix generated by the affine mapping N = 2n + 3. (b) Orthogonal lattice representation of the residue classes n ≡ i (mod 2i + 3). (c) Nested-square representation generated by the same arithmetic progressions. The three panels encode exactly the same divisibility infor- mation and differ only in their geometric visualization. 14.2. Orthogonal Lattice Representation. Each arithmetic progression may also be represented geometrically as an orthogonal lattice. Every collision corresponds to the intersection of a vertical and a horizontal segment associated with the same residue class. The resulting lattice provides a geometric visualization of the congruence structure encoded by the Collision Matrix while preserving exactly the same arithmetic information. 14.3. Nested Square Representation. The same residue classes admit a second geometric interpretation. For each integer m ≥ 0, the quantity L = i + m(2i + 3) may be interpreted as the side length of a square whose vertices are gen- erated by the corresponding arithmetic progression.
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  24 GIANLUCA REMIGIOPISANO The collection of all such squares produces a family of nested geometric configurations whose spatial organization reflects the underlying modular structure. Although these configurations frequently exhibit visually self-similar pat- terns, no claim of fractal structure is made in the present work. Proposition 14.1 (Equivalent Geometric Representations). For every di- visor index i ≥ 0, the congruence class n ≡ i (mod 2i + 3) admits four equivalent representations: (1) as a row of the Collision Matrix; (2) as an arithmetic progression in the affine index space; (3) as an orthogonal lattice generated by the corresponding residue class; (4) as a family of nested squares generated by the same progression. Each representation encodes exactly the same divisibility information and differs only in its geometric interpretation. Proof. The Collision Matrix is generated by the arithmetic progression n = i + m(2i + 3), m ≥ 0. Representing each progression as lattice intersections or as the side lengths of nested squares does not modify the underlying congruence relation. Since every point of each geometric construction corresponds uniquely to an element of the same arithmetic progression, all four representations are mathematically equivalent descriptions of the same set of divisible indices. □ 14.4. Discussion. These geometric representations should be regarded as alternative visualizations of the Collision Matrix rather than independent algorithms. Their principal purpose is to reveal the regularity of the congruence classes generated by the affine mapping N = 2n + 3, thereby providing an intuitive interpretation of the collision structure un- derlying the proposed primality criterion. Possible connections with self-similar geometric patterns or other lattice structures remain open questions for future investigation and are beyond the scope of the present work.
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  THE 2N+3 SIEVE25 15. Future Work Several directions remain open. Future work includes • a complete algebraic characterization of Collision Matrices; • asymptotic density estimates for collision distributions; • graph-theoretical representations; • sparse matrix implementations; • SIMD and GPU realizations; • extensions to integer factorization. 16. Conclusions We introduced a congruence-based representation of odd primality through the affine mapping f(n) = 2n + 3. Within this framework, divisibility becomes equivalent to the occurrence of periodic collisions inside an infinite binary matrix. The resulting Collision Matrix provides a compact algebraic description of odd divisibility and yields a deterministic primality criterion equivalent to the classical square-root test. The framework also reveals a regular periodic organization of divisibil- ity relations within the index space, suggesting several avenues for further mathematical investigation. Code Availability The Python implementation of the 2n+3 sieve and the interactive collision matrix generator are freely available on Google Colab at: https://colab. research.google.com/drive/140a7AUN-71pdKg7KWbdWb19fcEBtGa9m. The Python program for Geometric Analysis of the 2n+3 Sieve is avail- able on Google Colab at: https://colab.research.google.com/drive/ 18sS6OwywdRIQ9_vB5GkcP5REyUScNNqk. Acknowledgements During the preparation of this manuscript, the author used OpenAI’s ChatGPT (GPT-5.5) as a writing and editorial assistant to improve the clarity, organization, and presentation of the text and to assist in LaTeX typesetting. The author reviewed, verified, and takes full responsibility for all mathematical statements, proofs, algorithms, interpretations, and con- clusions contained in this article.
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