Locate the first error in the following derivation and give reasons for your answer:
Step A: A prime factor $p$ of $2^{61}-1$ is of the form
$$p= 122k + 1$$
Step B: With $k = 1$ we have $p = 123$ which is composite.
Step C: With $k = 2$ we have $p = 245$ which is composite.
Step D: With $k = 3$ we have $p = 367$ which is prime.
Step E: $p= 367 ≡ −1 \pmod 8$. Hence $p= 367$ is a prime factor of $2^{61} - 1$.
Step A is correct because 61 is an odd prime.
Steps B-D are correct numerically.
Step E is incorrect. Proposition (4.19)(a) says that a prime $p=2n+1$ for some integer $n$ is a factor of $2^n-1$ if $p \equiv \pm1 \pmod 8$. Here $367 \ne 2(61)+1$.