The board / number theory / #483
Erdős Problem #483
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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A monochromatic solution to \(a+b=c\) in \(\{1,\ldots,N\}\). Index \(i\) represents \(i+1\), and equal summands are allowed.
Every colouring of \(\{1,\ldots,N\}\) with at most \(k\) colours has a Schur triple.
Let \(f(k)\) be the minimal \(N\) such that if \(\{1,\ldots,N\}\) is \(k\)-coloured then there is a monochromatic solution to \(a+b=c\). Estimate \(f(k)\). In particular, is it true that \(f(k) < c^k\) for some constant \(c>0\)? The exponential bound is stated directly: for every positive number of colours, there is a forcing interval of length strictly below \(c^k\). This is equivalent to the bound on the least forcing threshold and avoids choosing a default value for that threshold.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/483.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA030126
Original pageerdosproblems.com/483
Attempts
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Explicitly reverified an 8-coloring of [1,5362], derived a checked infinite lower-bound subsequence, repaired the printed width-111 template, and proved the width-380 tail constant 148 locally maximal
PARTIAL