show that 31728 is congruent 1 mod 1729SolutionTo prove that

show that 3^1728 is congruent 1 (mod 1729)

Solution

To prove that 3¹⁷²⁸-1 = 1729k

First let us factorise 1729

1729 = 7x13x19

For each of prime factor p, p-1 divides 1728

Hence 3¹⁷²⁸ cannot be 0 mod 1729

Apply Fermat theorem

3¹⁷²⁸ =1 mod 7 and 3¹⁷²⁸ =1 mod 13, 3¹⁷²⁸ =1 mod 19.

By the Chinese remainder theorem it follows that

3¹⁷²⁸ =1 mod 1729 and that is, 1729 is a Carmichael number