The unit digit of the largest known prime

 

The main problem

 

          Find the unit digit of the largest prime number known:

                          213466917 – 1       (14 Nov, 2001 by Michael Cameron et al.)

This number contains 4,053,946 digits!

 

          Difficult? Let us investigate the unit digit of a power of natural number before we come back to this problem.

 

 

Period is 4

 

          First we like to define a function  u(n)  as the unit digit of the natural number n.

For example,         u(10) = 0

                                  u(1997) = 7

                                  u(25761992) = 2

 

We observe that :  u(2) = 2, u(22) = 4, u(23) = 8, u(24) = 6, u(25) = 2, u(26) = 4, u(27) = 8,….

So   u(2) = u(25),  and the pattern repeat itselves.

 

          We say that the period is 4, and the unit digit of the power of a natural number is the same after 4 numbers written in a sequence.

 

          Is it true for numbers other than 2? We like to prove that n and n5 have the same unit digit.

 

 

The Proof

 

       The proof is not difficult if we know some factorization:

 

               n5 – n     = n(n4 – 1) = n [(n2)2 – 1] = n(n2 –1)(n2 +1)

                               = n(n –1)(n + 1)(n2 +1)

                               = (n – 1)n(n + 1)[(n – 2)(n + 2) + 5]

                               = (n – 2)(n – 1)n (n + 1)(n + 2) + 5 (n – 1)n(n + 1)

 

       Now, five consecutive natural numbers  n – 2, n – 1, n, n + 1, n + 2  must contains a factor of 2 and a factor of 5, and their product must be divisible by 10.

       Similarly,  since (n – 1)n(n + 1) is divisible by 2,  5(n – 1)n(n + 1) is divisible by 10.

 

       Therefore n5 – n is divisible by 10.  The unit digit of  n5 – n  is 0.

Finally  u(n5 ) = u(n).

 

 

Mathematical Induction

 

       It is a good exercise to prove :

 

               “n5 – n is divisible by 10 , where n is a natural number.”

       using Principle of Mathematical Induction.

 

The calculations are not given here. There are some twists in the proof. Enjoy proving yourselves.

 

 

The main problem again

 

       Problem : Find  u(213466917 – 1).

 

Now,     13466917 = 3366729 ´ 4 + 1,          remember the period is 4.

       u(213466917 )   = u(23366729 ´ 4 + 1)

                               = u(21) ,                                  discard 23366729 ´ 4, it is repeating.

                               = 2

\ u(213466917 – 1) = 2 – 1 = 1

\ The unit digit of the largest known prime number 213466917 – 1 (up to May, 2003) is 1.

 

               If you like to have a look to this very large prime number and check the last digit (don’t print, it needs a lot of paper), chick here.