The largest known prime number

The largest known prime as of the 5th January 2015 is the 17,425,170 decimal digit Mersenne prime 2^57885161-1. Assuming that you could write at one digit a second, it would take you about six and a half months even to write this number down. I want to demonstrate the power of the exponentiation modulo n program that I wrote for the Casio fx-50F by calculating some of the last digits of this number.

Firstly, though, I will look at some Mersenne primes that we can write down relatively easily. M₃₁ can be computed on the fx50F and is the number 2³¹-1=2147483647. We can verify that the last digit is 7 by computing M₃₁ modulo 10. What we actually do is compute 2³¹ (mod 10) and subtract 1 (mod 10) which comes to the same thing. Running the exponentiation program with X=2, A=31 and B=10 gives the result D=8. That is 2³¹ (mod 10) is congruent to 8 (mod 10). Subtracting 1 (mod 10) from this gives us 7, which is the anticipated last digit of M₃₁.

M₁₂₇ is the number 2¹²⁷-1=170141183460469231731687303715884105727. Running the program with X=2, A=127 and B=10 gives the result D=8 from which we again deduce that the last digit is 7.

So what about M_57885161? Can the program compute the last decimal digit of such a large number? Running the program with X=2, A=57885161 and B=10 gives D=2 after 27 seconds of processing. This means that the last digit of this number is 1. We can also find the last two digits by computing M_57885161 (mod 100) and after 2 minutes 47 seconds the program calculates these to be 51. A further calculation (mod 1000) lasting 23 minutes shows that the last three digits are 951.

So what's the next largest Mersenne number? The next largest prime after 57885161 is 57885167 and so 2^57885167-1 may or may not be a prime. If it isn't prime then it will have a factor which is a number of the form 115770334n+1 where n is a positive integer. Good luck with trying out your long division!