For every $n > 2$ there exist distinct integers $1 \leq x < y < z$ such that $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} .$

Open problem — our best result is machine-sealed: verified reduction, reproduced by an independent verifier. The conjecture itself is unsettled.

number theory · open · formalized (Lean) · 1 attempt

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

verified reduction

needs verification

Transfer from #306: verified 4/n=1/x+1/y+1/z (1<=x<y<z) for every n=3..300, 0 failures (Opus re-checked 298 stored witnesses exactly).

#306 congruence-family habit in the stricter 3-term Erdos-Straus setting; reduces 4/n-1/x to (ay-b)(az-b)=b^2. SCOPE: exact finite verification + method transfer; NOT a proof of Erdos-Straus.

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

What you wrote is (essentially) the **Erdős–Straus conjecture**:

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_242 (n : ℕ) (hn : 2 < n) :
    ∃ x y z : ℕ, 1 ≤ x ∧ x < y ∧ y < z ∧
      (4 / n : ℚ) = 1 / x + 1 / y + 1 / z

formal-conjectures/242.lean ↗

oeis

A073101 — Number of integer solutions (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.0,0,1,1,2,5,5,6,4,9,7,15,4,14,33,22,4,21,9,30,25,22,19,45,10,17,25,36,7,72,17,62,27,22,59,69,9,29,67,84,7,77,12,56,87,39 A075245 — x-value of the solution (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The y and z components are in A075246 and A075247.1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10,11,11,11,11,12,12,12,12,13,14,13,13,14,14,A075246 — y-value of the solution (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The x and z components are in A075245 and A075247.4,3,4,7,15,7,10,16,34,13,18,29,61,21,30,46,96,31,43,67,139,43,60,92,190,57,78,121,249,73,100,154,316,91,124,191,391,111,A075247 — Largest possible z-value of an integer solution (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z. The x and y components are in A075245 and A075246.12,6,20,42,210,42,90,240,1122,156,468,812,3660,420,510,2070,9120,930,1806,4422,19182,1806,2100,8372,35910,3192,9048,1452 A075248 — Number of solutions (x,y,z) to 5/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.0,1,2,1,1,3,5,9,6,3,12,5,18,15,10,5,21,11,22,18,15,8,55,30,15,20,43,20,45,5,24,35,23,36,53,10,21,52,62,6,62,12,73,69,16,A287116 — Nonsquare integers that cannot be represented in the form 4M-d, where (a*b)|M and d|(a+b) for some positive integers a,b.288,336,4545

links

Erdős-Straus conjecture · reference

related: #306

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