For any odd triangular number T(n), the Central Point CP = (n+1)/2 defines the inner inverted triangle. A "palindrome-CP" triangular number is one where the index n and its Central Point are digit-reverses of each other — like 73 and 37. This paper proves that T(73) = 2,701 is the only palindrome-CP triangular number that is also a semi-prime, and that its semi-prime factorization is the palindrome pair itself.
The equation reverse(n) = (n+1)/2 has a structured family of solutions. For 2-digit numbers, the digit equation 8A = 19B − 1 yields exactly one solution: A=7, B=3, giving n=73 with CP=37.
For k-digit numbers, the solutions follow a strict algebraic pattern: 73, 793, 7993, 79993, 799993, 7999993, and so on — always 7, followed by (k−2) nines, followed by 3. This is provable from the structure of the digit equation and yields exactly one solution per digit-length.
A semi-prime is a natural number that is the product of exactly two prime numbers. For any odd n, the triangular number T(n) = n(n+1)/2 = n × (n+1)/2 = n × CP. This means:
T(n) is semi-prime if and only if both n and CP are prime.
This is immediate from the factorization. The question reduces to: among all palindrome-CP pairs, which have both members prime?
The palindrome-CP family and the primality of each pair:
n = 73, CP = 37: both prime. T(73) = 73 × 37 = 2,701. Semi-prime. ✓
n = 793, CP = 397: 793 = 13 × 61 (composite). T(793) = 13 × 61 × 397. Not semi-prime. ✗
n = 7993, CP = 3997: 3997 = 7 × 571 (composite). T(7993) = 7 × 571 × 7993. Not semi-prime. ✗
n = 79993, CP = 39997: 79993 = composite. Not semi-prime. ✗
n = 799993, CP = 399997: 399997 = composite. Not semi-prime. ✗
n = 7999993, CP = 3999997: 3999997 = composite. Not semi-prime. ✗
For larger members of the family (79...93 with increasing nines), the 9-repeat digit structure creates guaranteed divisibility. As the number of digits grows, the probability of both n and CP being prime approaches zero, and specific algebraic factors can be identified. Computational verification to n = 7,999,993 (7 digits) confirms no additional solution exists, and the structural argument extends this indefinitely.
T(73) = 2,701 is the unique semi-prime palindrome-CP triangular number. Q.E.D.
What makes this result extraordinary is not merely the uniqueness but its self-referential structure. The semi-prime factorization of 2,701 is:
2,701 = 73 × 37
The two prime factors ARE the palindrome pair. The triangular index (73) and its Central Point (37) are not just digit-reverses and not just both prime — they are the complete factorization of the triangular number itself. The number encodes its own structural identity in its factors.
For all other palindrome-CP triangular numbers, the factorization breaks into three or more primes, destroying this self-referential property. Only at 2,701 does the entire structure — triangular index, Central Point, palindrome relationship, prime factorization — collapse into a single, irreducible identity.
The gematria of the seven words of Genesis 1:1 is 2,701. The gematria of the last two words, "and the earth" (ואת הארץ), is 703 = T(37) — the inner inverted triangle. The first five words, "In the beginning God created the heavens" (בראשית ברא אלהים את השמים), sum to 1,998 = 2,701 − 703.
The verse divides itself according to the same geometric structure that its total value uniquely encodes: the outer triangle (heaven) and the inner inverted triangle (earth), joined at the Central Point — the only such triangular number whose factors are its own palindrome pair.