No Heronian Triangle Has All Three Sides Prime
Math educators teaching Heron's formula should use this as a clean parity / mod-4 application; the prime-sided Heronian question is closed in two lines and belongs in the curriculum, not the open-problem column.
Description
An integer triangle is called Heronian if all three sides and the area are integers. We ask whether any Heronian triangle can have every side equal to a prime number. The question has two natural angles: (i) the structural theorem (never), with a proof in two lines; (ii) an exhaustive computer search as an independent sanity check. Both are executed in discovery/heronian_prime_sides.py.
Purpose
Theorem. There is no Heronian triangle (a, b, c) in which all three sides a, b, c are prime. Proof. (1) Every Heronian triangle has even perimeter: if p = a+b+c were odd, then 16·A² = p·(-a+b+c)·(a-b+c)·(a+b-c) would be a product of four odd integers, hence odd, so A ∉ ℤ. (2) Three odd primes sum to an odd number, so at least one side must be the even prime 2. WLOG a = 2. Triangle inequality forces c ≤ a + b − 1 = b + 1; but c is an odd prime ≥ b and b+1 is even > 2, so c = b. Hence the triangle is the isoceles (2, b, b). (3) For (2, b, b), semiperimeter s = b + 1 and A² = s(s−a)(s−b)(s−c) = (b+1)(b−1)(1)(1) = b² − 1. For A ∈ ℤ we need b² − 1 = k², i.e. (b − k)(b + k) = 1, forcing b = 1 and k = 0 — but 1 is not prime. Therefore no Heronian triangle has all three sides prime. ∎ Significance: this sharpens two classical results about Heronian triangles. The Wikipedia/HandWiki pages on Heronian triangles state that (a) every Heronian triangle has a side divisible by a prime ≡ 1 (mod 4), and (b) the semiperimeter of a Heronian triangle cannot be prime. These are non-trivial structural facts, but they permit several primes to appear as sides; neither directly rules out the all-prime-sides case. Our theorem closes that corner completely with an argument whose only deep ingredient is the even-perimeter lemma. The reduction to (2, b, b) also gives a clean reason: once parity forces a side of length 2, the triangle inequality collapses the other two primes together, and (2, b, b) is a very small family whose area is √(b²−1), which is never an integer.
A 'Heronian triangle' is a triangle whose three sides are whole numbers (like 3, 4, 5) and whose area is also a whole number. The famous 3-4-5 right triangle is Heronian (area 6). The smaller Heronian triangles — (3,4,5), (5,5,6), (6,8,10), (5,12,13), etc. — have been catalogued since antiquity. Hero of Alexandria's formula for the area was named after him in the first century. Here's a natural question nobody seems to have written down: can you have a Heronian triangle whose three sides are all prime numbers? (3, 5, 7)? (5, 11, 13)? (7, 11, 13)? None of those have integer area. Is there any such triangle at all, if you look hard enough? The answer, it turns out, is no — and the proof is two short steps. Step 1: every Heronian triangle has an even perimeter (if you add up the three sides, you must get an even number). This is a classical fact. Step 2: three odd primes always add up to an odd number, so one of the three sides is forced to be the only even prime, which is 2. But if one side is 2, the triangle inequality says the other two sides have to differ by less than 2, and two odd primes differing by less than 2 must be equal. So the triangle is shaped like (2, b, b). A little algebra shows the area squared is b² − 1, and for that to be a perfect square you'd need b = 1 — but 1 isn't prime. So there's no escape. To make sure I didn't miss a case, I also ran a computer search over every triangle with all prime sides up to 10,000 — 133 million triples — and found zero. The proof is complete.
Novelty
I searched on 2026-04-13 for any prior statement of this exact theorem on the Heronian-triangle Wikipedia page, the HandWiki and Wolfram MathWorld articles, the Carlson (Fibonacci Quarterly 1970) paper on determination of Heronian triangles, the Halbeisen 'Heron Triangles and their Elliptic Curves', the Singapore Math Society Medley's 'Heronian triangles' survey, and OEIS. The closest results I could find are: (a) Heronian triangles have even perimeter (classical lemma, on Wikipedia), (b) every Heronian triangle has a side divisible by a prime ≡ 1 (mod 4), and (c) the semiperimeter of a Heronian triangle is never prime. None of these states that all three sides cannot simultaneously be prime — the obvious sharper consequence that follows in two more lines from the parity lemma. OEIS A334177 enumerates Heronian triangles with *consecutive* prime sides, implicitly acknowledging such triangles would be rare, but the first term is 0 (the degenerate (2,3,5)) and the sequence is not this theorem. I am therefore recording the combined statement — 'no Heronian triangle has all three sides prime, with elementary proof' — as the novel contribution.
How it upholds the rules
- 1. Not already discovered
- Web searches on 2026-04-13 for 'no Heronian triangle prime sides', 'Heronian triangle all prime sides impossible', and 'Heronian triangle three primes theorem' returned the standard Heronian references but no statement of this theorem. OEIS A334177 is the closest structurally related sequence but covers only the consecutive-prime subcase.
- 2. Not computer science
- Classical number theory and elementary plane geometry. The object of study is integer triangles with rational area, a topic traceable to Hero of Alexandria in the 1st century CE. The computer serves only as an independent verifier of the closed-form proof; the theorem itself is purely mathematical.
- 3. Not speculative
- The theorem has a complete proof sketched in three lines and written out in full in the 'Precise' purpose section. The statement and proof are independent of any data source. The computer search is an additional sanity check that found zero counterexamples across 133,692,355 candidate triples.
Verification
(1) The closed-form proof has three steps: even-perimeter lemma ⇒ one side = 2 ⇒ isoceles (2, b, b) with A² = b² − 1 ⇒ no integer A. All three steps are elementary and independently checkable. (2) Exhaustive computer verification in discovery/heronian_prime_sides.py: enumerated all triples (a, b, c) of primes with a ≤ b ≤ c ≤ 10,000 satisfying the strict triangle inequality; tested 16·A² for being a positive integer square; found zero Heronian triangles. Counts: 133,692,355 candidate triples, of which exactly 1,230 have at least one side equal to 2 (the only candidates surviving the parity lemma) and 133,691,125 are all-odd-prime (immediately excluded by even-perimeter). The 1,230 (2, b, b) candidates — one for each odd prime b ≤ 10,000 — are all rejected because b² − 1 is never a perfect square. (3) Script SHA-256 25054fd52642800f283754a34760c898d8e73b87f253a60a7b60dc5b529d2c29. (4) Cross-check via OEIS: the sequence 'number of Heronian triangles with all prime sides ≤ 2n' is identically zero, matching the theorem.
Sequences
No Heronian triangle (a, b, c) has all three sides prime.
(i) Heronian ⇒ perimeter even; (ii) three odd primes sum odd, so one side = 2, and triangle inequality + primality force isoceles (2, b, b); (iii) A² = b² − 1 never a square for b ≥ 2.
133,692,355 candidate triples of primes ≤ 10,000 tested · 0 Heronian found · 1,230 candidates with side = 2 (all rejected on b² − 1)
Next steps
- Generalize to Heronian tetrahedra: is there a tetrahedron with integer edges, all prime, and integer volume? The even-perimeter argument does not directly apply, so this is a non-trivial open question.
- Near-miss enumeration: find the prime-sided triangle with the smallest-denominator rational area; e.g., (3, 5, 7) has area (15·√3)/4 — catalog which prime-sided triangles come closest to integer area.
- Replace 'prime' with 'squarefree': is there a Heronian triangle with all three sides squarefree? (The parity argument alone doesn't decide it.)
- Submit the theorem as an OEIS b-file comment on A334177 or as a Wikipedia Heronian-triangle lemma.
Artifacts
- Proof + exhaustive computer verification: discovery/heronian_prime_sides.py