You will recall that a prime number is a number whose only factors are +/- itself an +/- 1. The largest known prime number to date was discovered on June 1, 1999, by members of the Great Internet Mersenne Prime Search. It is the number 2^6972593 - 1.
A) Without multiplying it out, determine how many digits are needed to write out this number in base-10.
B) Use the factor theorem to show that if 2^p -1, where p does not equal 3, is a prime number, then p is neither divisible by 4 or divisible by 3. [Alternatively, prove that if p is divisible by 4 or 3, then 2^p-1 is divisible by some number other than +/- itself or +/- 1.]
**This looks like a fun quesiton, but I have NO IDEA where to start!! Please help!!
A) Without multiplying it out, determine how many digits are needed to write out this number in base-10.
B) Use the factor theorem to show that if 2^p -1, where p does not equal 3, is a prime number, then p is neither divisible by 4 or divisible by 3. [Alternatively, prove that if p is divisible by 4 or 3, then 2^p-1 is divisible by some number other than +/- itself or +/- 1.]
**This looks like a fun quesiton, but I have NO IDEA where to start!! Please help!!