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It is shown that there are infinitely many positive integers  k  such that the congruence      2 n −  k ≡  1 ( mod n ) {2^{n - k}} \equiv 1\;\pmod n       has infinitely many solutions  n .

On the congruence 2^{𝑛-𝑘}≡1(𝑚𝑜𝑑𝑛)