Question

Activity 6.6.

(a)

A positive integer that is greater than 11 and not prime is called composite.

Write a technical definition for the concept of composite number with a similar level of detail as in the “more complete” definition of prime number.

Note.

A number is called prime if its only divisors are 1 and itself.

This definition has some hidden parts: a more complete definition would be as follows.

A number is called prime if

1. it is an integer,
2. it is strictly greater than 1,1, and
3. there does not exist any other number greater than 11 which divides it.

Don't just define it as “not prime.” And make sure that the equality 7=1×7 can't be used to justify the statement “7 is composite” by your definition (because prime 7 is most definitely not composite).

(b)

Prove by proving the contrapositive: If 2^n−1 is prime, then n is prime.

hint:

You may find the following “difference of powers” factorization formula useful:

a^m−b^m=(a−b)(a^m−1+a^m−2b+a^m−3 b^2+⋯+a^2b^m−3+ab^m−2+b^m−1).

Hint

Answer 1