Problem 850

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claims/: The 1 claim page of Problem 850, one per claimant's result; the problem's standing derives from them.

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Statement. 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?

Formulation. The site leaves the range of $x$ and $y$ implicit. Its curator states in the problem's forum thread (19 January 2026) that $x,y\ge1$ is meant. The formal-conjectures statement takes $x$ and $y$ to be natural numbers, and zero adds no solution. Over all integers the question has the trivial answer yes ($x=2$, $y=-4$: the pairs $2,-4$, $3,-3$ and $4,-2$ each have the same prime factors), but that is not the problem asked. This page reads the problem for positive integers.

Status. Open.

Source. erdosproblems.com/850, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #850, https://www.erdosproblems.com/850.

References.

• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp. Section B29 "Is $x$ determined by the prime divisors of $x+1$, $x+2$, ..., $x+k$?", printed p. 127: Woods's question, "Perhaps $k=3$?", the four ambiguous cases for $k=2$ in which only primes below 23 occur, the largest being $(x+1,x+2)=(75,76)$ or $(1215,1216)$, and the infinite family $(2^n-2,2^n-1)$, $(2^n(2^n-2),(2^n-1)^2)$; B19 "$(m,n+1)$ and $(m+1,n)$ with same set of prime factors. The abc-conjecture.", printed p. 113, has Erdős's two-term question with Mąkowski's pair $m=3\cdot5^2$, $n=3^5\cdot5$. Library home: guy_2004_unsolved_problems_number_theory.
• [Ma68] Makowski, Andrzej, On a problem of Erdős. Enseign. Math. (2) (1968), 193.
• [ShTi16] Shorey, Tarlok N. and Tijdeman, Rob, Arithmetic properties of blocks of consecutive integers. (2016), 455-471.

Formalization. Statement in formal-conjectures.

Current assessment

The status above is imported from the dated site record. Shorey and Tijdeman prove that the answer is no under Baker's explicit form of the abc conjecture; their claim page records that conditional result, which gives no unconditional answer. The note below records an author-recorded reconstruction from a research folder and is not independently reviewed. This page records no current literature search or independent assessment of proof coverage.

Known Results

Lemma 3.2 of Pollack, Pomerance and Treviño, reconstructed on its page, bounds the number of $j$ with $j$ and $j+k$ sharing their prime factors by $3\cdot7^{3+2\omega(k)}$ for each fixed $k$.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• guy_2004_unsolved_problems_number_theory
• shorey_2016_arithmetic_properties_blocks_consecutive_integers
• shorey_2016_arithmetic_properties_blocks_consecutive_integers / theorem_9_1