Can there exist two distinct integers $x$ and $y$ such that $x, y$ have the same prime factors, $x + 1, y + 1$ have the same prime factors, and $x + 2, y + 2$ also have the same prime factors?

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

Interpreting “prime factors of an integer” as the primes dividing its absolute value, take [ x=-4,\qquad y=2. ]

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_850 :
    answer(sorry) ↔ ∃ x y : ℕ, x ≠ y ∧ x.primeFactors = y.primeFactors
      ∧ (x + 1).primeFactors = (y + 1).primeFactors
      ∧ (x + 2).primeFactors = (y + 2).primeFactors

formal-conjectures/850.lean ↗

oeis

A343101 — Pairs of integers (k, m) ordered by m with 1 < k < m such that k has the same prime divisors as m, and, k+1 has the same prime divisors as m+1.2,8,6,48,14,224,30,960,75,1215,62,3968,126,16128,254,65024,510,261120,1022,1046528,2046,4190208,4094,16769024,8190,67092

links

ABC conjecture · reference

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