Geometric Infinite Expansion of Squares and the Fractal Model of Prime Creation at the +1 Boundary
This paper introduces a geometric framework that defines numbers not as dimensionless points,
but as the areas of squares, emphasizing dimension and spatial extension. By modeling the interaction between
the infinity of the integer space and the self-similar (fractal) expansion of squares.
This is my formula which generate prime numbers.
In geometric terms, this formula is rigorously illustrated as follows:
1. Core Formation: A closed square domain of area p^2 , with a side length equal to the prime p.
2. Asymmetric Expansion (Gnomon): Two rectangular domains of width (d−1) and length p, structurally appended to the horizontal and vertical boundaries, denoted by 2p(d−1). This process represents an infinite expansion algorithm wherein the square consumes the external integer space while strictly preserving its own self-similarity.
Squares possess the highest structural affinity for fractals due to their capacity for infinite grid-like division and consolidation. As d progresses toward infinity (d → ∞), the macroscopic extension of space continuously replicates the microscopic grid structure of p^2 . Therefore, as long as the integer space exists, this self-similar geometric expansion remains unstoppable.
A square is inherently a "closed shape"—a finite harmonic domain enclosed by four edges. The total area generated by the formula p^2 + 2p(d−1) represents a fully governed "fortress of composite numbers," dictated entirely by the factor p. However, no matter how infinitely this fortress expands, the very first grid unit immediately outside the finalized domain is always invariant, expressed mathematically as +1.
Even as the spatial scale approaches infinity, the fundamental property of the minimal unit "+1" neither shrinks nor vanishes; it remains completely intact ("the plus one remains as it is"). Geometrically, this +1 domain—positioned just a single step outside the vertex (corner) of the square— becomes an independent, invariant region, completely immune to the mathematical waves of any preexisting factors or multiples. This critical point (such as p^2 + 1), entirely isolated from the surrounding harmony, induces a localized breaking of number-theoretic symmetry. It is precisely at this point that a new prime number is inevitably manifested and suddenly brought into existence.
Legendre's conjecture asserts that there always exists at least one prime number between n^2 and (n+1)^ 2 .
According to the model presented in this paper, the conjecture is resolved as follows:
Given the vast canvas of the infinite integer space, a square never remains permanently closed; it continuously
shifts and expands outward in pursuit of the next +1. As the square transitions to the next dimension (the next
larger square), a structural "gap" is automatically generated at its critical boundary line (the +1 threshold),
which can never be invaded by other factors.
Paralleling this, it is proven that the very movement of the infinite expansion of the integer space—mediated
by the invariant remainder +1—is the perpetual driving force that continuously yields prime numbers into the
furthest reaches of infinity.
1. Core Formation: A closed square domain of area p^2 , with a side length equal to the prime p.
2. Asymmetric Expansion (Gnomon): Two rectangular domains of width (d−1) and length p, structurally appended to the horizontal and vertical boundaries, denoted by 2p(d−1). This process represents an infinite expansion algorithm wherein the square consumes the external integer space while strictly preserving its own self-similarity.
Squares possess the highest structural affinity for fractals due to their capacity for infinite grid-like division and consolidation. As d progresses toward infinity (d → ∞), the macroscopic extension of space continuously replicates the microscopic grid structure of p^2 . Therefore, as long as the integer space exists, this self-similar geometric expansion remains unstoppable.
A square is inherently a "closed shape"—a finite harmonic domain enclosed by four edges. The total area generated by the formula p^2 + 2p(d−1) represents a fully governed "fortress of composite numbers," dictated entirely by the factor p. However, no matter how infinitely this fortress expands, the very first grid unit immediately outside the finalized domain is always invariant, expressed mathematically as +1.
Even as the spatial scale approaches infinity, the fundamental property of the minimal unit "+1" neither shrinks nor vanishes; it remains completely intact ("the plus one remains as it is"). Geometrically, this +1 domain—positioned just a single step outside the vertex (corner) of the square— becomes an independent, invariant region, completely immune to the mathematical waves of any preexisting factors or multiples. This critical point (such as p^2 + 1), entirely isolated from the surrounding harmony, induces a localized breaking of number-theoretic symmetry. It is precisely at this point that a new prime number is inevitably manifested and suddenly brought into existence.
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