For example, p = 17 is the lowest prime that satisfies both p == 1 (mod 8) and p == 1 (mod 16).
I guess the whole point of my question is that:
Is there a prime number satisfying p == 1 (mod 8) but NOT p == 1 (mod 16)?
The original question was ill-posed.
Indeed, p = 41 is a counterexample:
p = 41 == 1 (mod 8)
while
p = 41 == 9 (mod 16)
The answer is no.
Starting from p prime and p 1==1 (mod8), it comes p==1 (mo16) or p== -7 (mod16). But the occurrences for a prime number modulo 16 are p== +-1, +-3, +-5, +-7 (mod16). And the distribution of prime numbers is identical in each of these 8 classes (Dirichlet). So the prime numbers ==1 (mod8) cannot be reduced to those ==1 (mod 16) otherwise the class of numbers == -7 (mod16) would be empty which is absurd example 41.
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