Question

21. Show that 229 - 1 is composite by doing it the way Fermat could have.

Answer 1

By Fermat's theorem, we know that, a^b -1 identicalto 1 (mod p) where p is a prime & a > 0 is an integer and p times a Here let a = 2 & p = 29 therefore 2^28 identicalto 1(mod 29) 2^29 identicalto 2(mod 29) 2^29 -1 identicalto 1(mod 29) 2^29 -1 = 29k + 1 for k elementof Z Now we know that if 2^n -1 is prime then n is also prime i.e., if n is composite then 2^n -1 is also composite. But here 29 is not composite Hence 2^29 -1 is not prime Thus 2^29 -1 is composite.