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Aug 5, 2019 at 17:53 comment added GH from MO It is also worthwhile to note that $n\mid b^n-1$ implies that $n$ has a common prime divisor with $b-1$.
Aug 5, 2019 at 15:36 comment added Max Alekseyev @HenriCohen: This would not help as $n=b-1$ still starts an infinite series of exceptional values in this case for any $b>2$ (notice that $(b-1)^2\mid (b^{b-1}-1)$).
Aug 5, 2019 at 11:59 comment added Henri Cohen Perhaps one could modify the OP's question to $n\nmid(b^n-1)/(b-1)$ ?
Aug 5, 2019 at 10:58 vote accept Dominic van der Zypen
Aug 5, 2019 at 10:46 history answered Max Alekseyev CC BY-SA 4.0