[seqfan] Question and Problem

[Tomasz Ordowski]

Hello SeqFans!

Let's define: Squarefree composite numbers k such that p-1 divides k-1 and
p-1 does not divide (k-1)/2 for every prime p|k.
These are Carmichael numbers k such that p-1 does not divide (k-1)/2 for
every prime p|k.
8911, 29341, 314821, 410041, 1024651, 1152271, 5481451, 10267951, 14913991,
15247621, 36765901, 64377991, 67902031, 133800661, 139952671, 178482151,
188516329, ... (*)
Question: Are these odd composite numbers k such that b^{(k-1)/2} == -1
(mod k) for some base b such that ord_{k}(b) = lambda(k)? How to prove the
equivalence of these definitions?

Problem: Are there odd composite numbers m such that b^{(m-1)/2} == -1 (mod
m) for all bases b such that ord_{m}(b) = lambda(m)? Of course, this is a
proper subset (maybe empty) of Carmichael numbers. Amiram Eldar did not
find any such number m < 2^64. How to prove that they do not exist?

Note that if p is an odd prime, then b^{(p-1)/2} == -1 (mod p) for all
bases b such that ord_{p}(b) = lambda(p) = p-1.

Best regards,

Thomas
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See also https://oeis.org/history/view?seq=A329538&v=19 a subset of (*).