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

- it is an integer,
- it is strictly greater than 1,1, and
- 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

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