A Unique Primitive Pythagorean Quadruple With All Four Sides Triangular
Number-theory practitioners enumerating primitive Pythagorean quadruples under arithmetic-set constraints should treat the triangular-numbers case as 'one solution exists, none others to T_999' — the search is closed.
Description
A Pythagorean quadruple is a 4-tuple (a, b, c, d) of positive integers with a² + b² + c² = d², i.e., an integer rectangular box (a × b × c) with integer space diagonal d. This is the 3D analogue of a Pythagorean triple. In iteration 48 we showed that a 2D triangular-sided Pythagorean triple exists uniquely (T_132, T_143, T_164) up to T_44720. Here we ask the 3D analogue: does any integer rectangular box with triangular-number edges have a triangular-number space diagonal?
Purpose
Enumeration result. Iterating every triple (T_i, T_j, T_k) with i ≤ j ≤ k ≤ 999 (so max side T_999 = 499,500) and checking whether T_i² + T_j² + T_k² is a perfect square AND lies in the triangular-number set yields exactly five Pythagorean quadruples with all-triangular components, sorted by hypotenuse d: (21, 120, 120; 171), (171, 210, 378; 465), (630, 1035, 1770; 2145), (10296, 37128, 71253; 81003), and (19110, 132355, 173166; 218791). The corresponding triangular indices are (T_6, T_15, T_15; T_18), (T_18, T_20, T_27; T_30), (T_35, T_45, T_59; T_65), (T_143, T_272, T_377; T_402), and (T_195, T_514, T_588; T_661). Of the five, gcd(a, b, c, d) equals 3, 3, 15, 39, and 1 respectively, so exactly one is primitive: (19110, 132355, 173166, 218791). Reduction of the non-primitives: the first factors as 3 × (7, 40, 40; 57), the second as 3 × (57, 70, 126; 155), the third as 15 × (42, 69, 118; 143), the fourth as 39 × (264, 952, 1827; 2077). Each reduced 4-tuple is itself a primitive Pythagorean quadruple that happens, under the right scaling, to land on triangular-number sides — a mechanism exactly analogous to the triangular-Pythagorean-triple situation in iteration 48 (where 66 × (133, 156, 205) = (8778, 10296, 13530)). The uniquely primitive solution (T_195, T_514, T_588, T_661), in contrast, is not obtained by scaling any smaller primitive Pythagorean quadruple onto triangular sides — it is intrinsically triangular. Verification: 19110² + 132355² + 173166² = 365,192,100 + 17,517,846,025 + 29,986,463,556 = 47,869,501,681 = 218,791². The indices decompose as 195 = 3·5·13, 514 = 2·257, 588 = 2²·3·7², 661 prime; and the sides 19110 = 2·3·5·7²·13, 132355 = 5·103·257, 173166 = 2·3·7·7·19·31 (not fully verified — 7² is in 19110, not 173166, but 173166 = 2·3·28861 = 2·3·7·4123 = 2·3·7·7·589 = 2·3·7²·589; 589 = 19·31). The complete lack of common factors across the four numbers was verified computationally. The significance is that (a) the 3D analogue of iteration 48 has 5 solutions rather than 1, giving a richer ledger, but (b) only one is primitive — a genuine singleton under the tightest filter — and (c) both the 4-tuple of sides and the 4-tuple of triangular indices are OEIS-absent as of 2026-04-13.
A Pythagorean triple is three whole numbers like (3, 4, 5) where the first two squared plus each other equal the third squared. A Pythagorean quadruple is the 3D version: four whole numbers (a, b, c, d) where a² + b² + c² = d². Geometrically, (a, b, c) are the three edges of a box, and d is the length of the straight line from one corner to the opposite corner through the middle of the box. The smallest Pythagorean quadruple is (1, 2, 2, 3): 1² + 2² + 2² = 9 = 3². They're not as famous as Pythagorean triples but they're well-studied. A triangular number is 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ... — the dots in a triangular arrangement. Question: is there any Pythagorean quadruple — any integer box — whose three edges AND space diagonal are all triangular numbers? The answer turns out to be: yes, but barely. Searching every box with sides at most the 999th triangular number (which is 499,500) finds exactly five such boxes. Four of them are 'boring' in a specific sense: they're smaller primitive Pythagorean quadruples multiplied by a scaling factor that happened to land on triangular numbers. Only one of the five is a genuinely primitive solution — a Pythagorean quadruple whose four sides share no common factor. That singleton is (19110, 132355, 173166, 218791), which are the 195th, 514th, 588th, and 661st triangular numbers respectively. You can check it: 19110² + 132355² + 173166² = 47,869,501,681 = 218791² exactly. I also checked OEIS (the big database of integer sequences), and neither the four sides nor the four triangular indices appear as any known sequence — so this singleton is apparently not previously catalogued. For context, iteration 48 of this project found exactly one 2D (triangle) Pythagorean triangular triple up to a billion, and zero primitives. Going to three dimensions unlocks exactly one primitive. The 3D version is somehow 'just barely' non-empty.
Novelty
OEIS queries on 2026-04-13 for both seq:19110,132355,173166,218791 and seq:195,514,588,661 returned 'No results.' The Pythagorean-quadruple Wikipedia article and Mathworld do not mention the triangular-number subcase. No prior source I could find catalogs the five all-triangular-sided Pythagorean quadruples up to T_999, let alone isolates the unique primitive (T_195, T_514, T_588, T_661).
How it upholds the rules
- 1. Not already discovered
- OEIS subsequence queries on both the side 4-tuple and the index 4-tuple returned 'No results.' Pythagorean-quadruple literature (Spira 1962, Oliverio, Diaz-Vargas et al.) discusses parametrizations and primitive-quadruple counts but not the triangular subcase.
- 2. Not computer science
- Classical number theory and integer 3D geometry. The object is a rectangular box with integer edges and diagonal. The computer enumerates over a finite range as an independent verifier.
- 3. Not speculative
- Every quadruple is reproducible by running discovery/pythagorean_quad_triangular.py. The bound T_999 = 499,500 is explicit and the 5-element list is complete under it.
Verification
(1) Brute-force over 166 million (i, j, k) triples with i ≤ j ≤ k ≤ 999 produces exactly 5 quadruples. (2) Manual check on the primitive: 19110² = 365,192,100; 132355² = 17,517,846,025; 173166² = 29,986,463,556; sum = 47,869,501,681 = 218,791². (3) gcd check: gcd(19110, 132355, 173166, 218791) = 1, confirming primitivity. (4) OEIS subsequence queries on both (19110, 132355, 173166, 218791) and (195, 514, 588, 661) return 'No results.' (5) Non-primitive reductions check out: 3·(7, 40, 40, 57) = first, 3·(57, 70, 126, 155) = second, 15·(42, 69, 118, 143) = third, 39·(264, 952, 1827, 2077) = fourth, each itself a primitive Pythagorean quadruple.
Sequences
(19110, 132355, 173166, 218791) = (T_195, T_514, T_588, T_661)
(T_6, T_15, T_15; T_18) = 3×(7,40,40;57) · (T_18, T_20, T_27; T_30) = 3×(57,70,126;155) · (T_35, T_45, T_59; T_65) = 15×(42,69,118;143) · (T_143, T_272, T_377; T_402) = 39×(264,952,1827;2077) · (T_195, T_514, T_588; T_661) PRIMITIVE · (T_899, T_1007, T_1519; T_1627) = 2×(202275, 253764, 577220; 662189)
19110² + 132355² + 173166² = 365,192,100 + 17,517,846,025 + 29,986,463,556 = 47,869,501,681 = 218,791² · gcd(a,b,c,d) = 1
Next steps
- Push the enumeration to T_5000 or T_10000 to determine whether (T_195, T_514, T_588, T_661) remains the unique primitive triangular Pythagorean quadruple, or whether a second primitive emerges at larger scale.
- Analyze the unique primitive structurally: does the 4-tuple (195, 514, 588, 661) fit any algebraic parametrization of primitive Pythagorean quadruples?
- Extend to 4D: Pythagorean 5-tuples (a² + b² + c² + d² = e²) with all five components triangular. Higher dimensions should eventually admit dense families.
- Submit the singleton to OEIS as a new sequence with keywords 'nice, fini, full' and cross-reference A000217 and A096907.
Artifacts
- Enumeration script: discovery/pythagorean_quad_triangular.py