The n conjecture is a conjecture in number theory stated by Jerzy Browkin and Juliusz Brzezinski in 1994 as a generalization of the abc conjecture to more than three integers.
Facts
StatementFor n at least 3, given nonzero coprime integers a1 through an that sum to zero and whose proper subsums are all nonzero, the n conjecture states that for every epsilon greater than 0 there is a constant depending on n and epsilon such that the maximum absolute value among the ai is less than that constant times the radical of the product of the ai, raised to the power 2n minus 5 plus epsilon. 1 Proposed Year Progress Toward ResolutionThe conjecture remains open. Work by Holzl, Kleine and Stephan in 2025 established that for n at least 5 the relevant limit superior is at least 5 over 3 for odd n and at least 5 over 4 for even n, while the cases n equal 3 and n equal 4 still lack nontrivial lower bounds and no common upper bound across all n has been shown. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source N Conjecture (Wikipedia)
Sources
1. N Conjecture (Wikipedia)
Lead section, definition
stated by Browkin & Brzezinski (1994) as a generalization of the abc conjecture
Lead section, attribution
stated by Browkin & Brzezinski (1994)
Recent progress section
for n >= 5 the above limit superior is for odd n at least 5/3 and for even n is at least 5/4
- Lead section
- In Branch: Number Theory, Lead sentence
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