Saturday, 24 January 2026

Exercise (4.4).8

Determine the first error in the following derivation and give reasons for your answer:

Step A: A prime factor $q$ of $2^{49} − 1$ is of the form

$$q= (2 × 49 × k) + 1 = 98k + 1$$

Step B: Substituting $k= 1$ into this $q= 98k + 1$ gives $q= 99$ which is composite.

Step C: Substituting $k= 2$ into this $q= 98k + 1$ gives $q= 197$ which is prime.

Step D: Therefore, 197 is a prime factor of $2^{49}− 1$.


Step A is incorrect.

Proposition (4.23) requires the index of 2 to be an odd prime. Here 49 is not a prime.