2 problems

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Odd-length linear equations are uncommon over the integers
Odd-length uncommonness conjecture. The equation is uncommon over the integers.
Conjectureopenadditive-combinatorics integer-colorings linear-equationsarXiv posted 2 days ago
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Costello–Elvin commonness conjecture for linear equations over the integers
Costello–Elvin's conjecture. The equation is common over the integers if and only if $k$ is even and $a_1,\ldots,a_k$ can be partitioned into pairs whose sums are zero.
Conjectureopenadditive-combinatorics integer-colorings linear-equationsarXiv posted 2 days ago

Odd-length linear equations are uncommon over the integers