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n conjecture
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<h2 id="formulations">Formulations</h2> <p>Given n ≥ 3 {\displaystyle n\geq 3} , let a 1 , a 2 , . . . , a n ∈ Z {\displaystyle a_{1},a_{2},...,a_{n}\in \mathbb {Z} } satisfy three conditions: </p> (i) gcd ( a 1 , a 2 , . . . , a n ) = 1 {\displaystyle \gcd(a_{1},a_{2},...,a_{n})=1} (ii) a 1 + a 2 + . . . + a n = 0 {\displaystyle a_{1}+a_{2}+...+a_{n}=0} (iii) no proper subsum of a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} equals 0 {\displaystyle 0} <p>First formulation </p><p>The <i>n</i> conjecture states that for every ε > 0 {\displaystyle \varepsilon >0} , there is a constant C {\displaystyle C} depending on n {\displaystyle n} and ε {\displaystyle \varepsilon } , such that: </p> <blockquote><p> max ( | a 1 | , | a 2 | , . . . , | a n | ) < C n , ε rad ( | a 1 | ⋅ | a 2 | ⋅ … ⋅ | a n | ) 2 n − 5 + ε {\displaystyle \operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|)<C_{n,\varepsilon }\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot \ldots \cdot |a_{n}|)^{2n-5+\varepsilon }} </p></blockquote> <p>where rad ( m ) {\displaystyle \operatorname {rad} (m)} denotes the <a href="/facts/Radical_of_an_integer/xyoLHVNs">radical</a> of an integer m {\displaystyle m} , defined as the product of the distinct <a href="/facts/Prime_factor/L20FLkgn">prime factors</a> of m {\displaystyle m} . </p><p>Second formulation </p><p>Define the <i>quality</i> of a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} as </p> q ( a 1 , a 2 , . . . , a n ) = log ( max ( | a 1 | , | a 2 | , . . . , | a n | ) ) log ( rad ( | a 1 | ⋅ | a 2 | ⋅ . . . ⋅ | a n | ) ) {\displaystyle q(a_{1},a_{2},...,a_{n})={\frac {\log(\operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|))}{\log(\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot ...\cdot |a_{n}|))}}} <p>The <i>n</i> conjecture states that lim sup q ( a 1 , a 2 , . . . , a n ) = 2 n − 5 {\displaystyle \limsup q(a_{1},a_{2},...,a_{n})=2n-5} . </p> <h2 id="stronger-form">Stronger form</h2> <p>Vojta (1998) proposed a stronger variant of the <i>n</i> conjecture, where setwise <a href="/facts/Coprime/bnCxCXxs">coprimeness</a> of a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} is replaced by pairwise coprimeness of a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} . </p><p>There are two different formulations of this <i>strong n</i> conjecture. </p><p>Given n ≥ 3 {\displaystyle n\geq 3} , let a 1 , a 2 , . . . , a n ∈ Z {\displaystyle a_{1},a_{2},...,a_{n}\in \mathbb {Z} } satisfy three conditions: </p> (i) a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} are pairwise coprime (ii) a 1 + a 2 + . . . + a n = 0 {\displaystyle a_{1}+a_{2}+...+a_{n}=0} (iii) no proper subsum of a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} equals 0 {\displaystyle 0} <p>First formulation </p><p>The <i>strong n</i> conjecture states that for every ε > 0 {\displaystyle \varepsilon >0} , there is a constant C {\displaystyle C} depending on n {\displaystyle n} and ε {\displaystyle \varepsilon } , such that: </p> <blockquote><p> max ( | a 1 | , | a 2 | , . . . , | a n | ) < C n , ε rad ( | a 1 | ⋅ | a 2 | ⋅ … ⋅ | a n | ) 1 + ε {\displaystyle \operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|)<C_{n,\varepsilon }\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot \ldots \cdot |a_{n}|)^{1+\varepsilon }} </p></blockquote> <p>Second formulation </p><p>Define the <i>quality</i> of a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} as </p> q ( a 1 , a 2 , . . . , a n ) = log ( max ( | a 1 | , | a 2 | , . . . , | a n | ) ) log ( rad ( | a 1 | ⋅ | a 2 | ⋅ . . . ⋅ | a n | ) ) {\displaystyle q(a_{1},a_{2},...,a_{n})={\frac {\log(\operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|))}{\log(\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot ...\cdot |a_{n}|))}}} <p>The <i>strong n</i> conjecture states that lim sup q ( a 1 , a 2 , . . . , a n ) = 1 {\displaystyle \limsup q(a_{1},a_{2},...,a_{n})=1} . </p><p>Hölzl, Kleine and Stephan (2025) harvtxt error: no target: CITEREFHölzl,_Kleine_and_Stephan2025 (help) have shown that for n ≥ 5 {\displaystyle n\geq 5} the above limit superior is for odd n {\displaystyle n} at least 5 / 3 {\displaystyle 5/3} and for even n {\displaystyle n} is at least 5 / 4 {\displaystyle 5/4} . For the cases n = 3 {\displaystyle n=3} (abc-conjecture) and n = 4 {\displaystyle n=4} , they did not find any nontrivial lower bounds. It is also open whether there is a common constant upper bound above the limit superiors for all n ≥ 3 {\displaystyle n\geq 3} . For the exact status of the case n = 3 {\displaystyle n=3} see the article on the <a href="/facts/Abc_conjecture/yJeHld4Q">abc conjecture</a>. </p> <ul><li><a href="/facts/Jerzy_Browkin/JQh5G7LL">Browkin, Jerzy</a>; Brzeziński, Juliusz (1994). "Some remarks on the <i>abc</i>-conjecture". <i>Math. Comp</i>. 62 (206): 931–939. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1994MaCom..62..931B">1994MaCom..62..931B</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2153551">10.2307/2153551</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2153551">2153551</a>.</li></ul> <ul><li>Hölzl, Rupert; Kleine, Sören; Stephan, Frank (2025). <a href="https://doi.org/10.1017%2FS1446788725000084">"Improved lower bounds for strong n-conjectures"</a>. <i>Journal of the Australian Mathematical Society (Accepted Paper on Journal Homepage)</i>. Australian Mathematical Society: 1–21. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FS1446788725000084">10.1017/S1446788725000084</a>.{{cite journal}}: CS1 maint: multiple names: authors list (link)</li></ul> <ul><li>Vojta, Paul (1998). <a href="https://doi.org/10.1155%2FS1073792898000658">"A more general abc conjecture"</a>. <i>International Mathematics Research Notices</i>. 1998 (21): 1103–1116. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/9806171">math/9806171</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1155%2FS1073792898000658">10.1155/S1073792898000658</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1663215">1663215</a>.</li></ul>