tricky prime number question

KarlyD

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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!!
 
For the first one, try using \(\displaystyle \L\\6972593log(2)\)

The whole number part is the the number of digits.


BTW, that number has been beaten since 1999. I believe it's close to 10 million digits now. The first person to find a prime over 10 million digits wins $50,000.
 
KarlyD said:
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.]
Hint:

Use p = 3n or 4m

Then use the fact that

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

and

a^4 - b^4 = (a-b)(a+b)(a^2 + b^2a)

**This looks like a fun quesiton, but I have NO IDEA where to start!! Please help!!
 
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