Mathematical Singularity of 2,701 — Part I
The Central Point Decomposition Singularity
Among all triangular numbers, 2,701 — the gematria of Genesis 1:1 — is the only one where the Hebrew numeral decomposition of the number equals the count of dots in its inner inverted triangle. Specifically: when 2,701 is written in the traditional Hebrew numeral system as ב׳תש״א (2 + 701), the sum 703 is identical to T(37), the inverted sub-triangle defined by the Central Point of T(73). This paper proves that this convergence is unique — it occurs for no other triangular number.
The proof requires establishing four conditions and then demonstrating that they can only be simultaneously satisfied at n = 73.
Condition 1: 2,701 = T(73)
A triangular number T(n) is the sum of the first n positive integers: T(n) = n(n+1)/2. Direct computation yields T(73) = 73 × 74 / 2 = 2,701. The index 73 is itself remarkable: it is prime, an emirp (its reverse 37 is also prime), and a star number.
Condition 2: The Central Point and the Inner Inverted Triangle
Every triangular number T(n) with odd index n possesses a Central Point (CP) — the geometric center of the triangular arrangement. For T(n), the Central Point is located at position (n+1)/2. For T(73), the Central Point is:
CP(73) = (73 + 1) / 2 = 37
This Central Point is not merely a position — it defines the inner structure. When 2,701 units are arranged as an equilateral triangle with 73 rows, the lower portion from row 37 to row 73 forms an inverted equilateral sub-triangle whose side length equals the Central Point itself. This inverted triangle contains exactly T(37) units:
T(CP) = T(37) = 37 × 38 / 2 = 703
The number 703 is the gematria of the final two words of Genesis 1:1: ואת הארץ ("and the earth"). The upper triangle and the inverted inner triangle correspond to "the heavens" and "the earth" — geometrically nested within the verse's total value.
Condition 3: The Hebrew Numeral Decomposition
In the traditional Hebrew numeral system, numbers above one thousand are expressed by separating the thousands from the remainder. The thousands are denoted by a letter with a geresh (׳), and the remainder is written as standard gematria. For example, the Hebrew year 5786 is written ה׳תשפ״ו: ה (= 5) represents the five-thousandth millennium and תשפ״ו (= 786) the year within it. The combined gematria is 5 + 786 = 791.
Applying this to 2,701: the Hebrew representation is ב׳תש״א, where ב (= 2) marks the second thousand and תש״א (= 701) is the remainder. The decomposed gematria is:
2 + 701 = 703 = T(37) = T(CP)
The total value of the first verse, when read through its own numeral system, collapses to the value of its geometric interior — the inverted triangle defined by its Central Point.
The Proof
We prove that 2,701 is the only triangular number for which the Hebrew numeral decomposition equals the inner inverted triangle.