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(Update, July 21, 2026): For $N<100$ and using a higher search radius, Simon Goater found,
while Jeff Braun found (June 27, 2026),
Guruprasad's earlier search radius of $|x|<20000$ found $6$ (and more) like,

Gardam [1] disproved the Kaplansky unit conjecture by exhibiting a nontrivial unit $\alpha$ of support size 21 in $\mathbb{F}_2[P]$, where $P = \langle a,b \mid b^{-1}a^2b = a^{-2},\ a^{-1}b^2a = b^{-2}\rangle$ is the Promislow group, and remarks that...

So as you may know, the Jacobian conjecture turned out to be false for polynomials in three or more variables. It holds in one variable, but the simplest non trivial case is that of two variables. Are there any reasons that point that It could it be...

This question is a copy of one I asked in the Math StackExchange forum a few days ago. I don't know if it qualifies as a research-level question, but it may be something beyond most people on the Math StackExchange forum.
Back in 2016, in his proposed answer to the question asked at https://math.stackexchange.com/questions/1790672/packing-of-n-balls/1795044#1795044 , achille hui ( https://math.stackexchange.com/users/59379/achille-hui ) gave the following definition of...