+130071 I believe that your teacher is having some fun with you. The first thing is one of the greatest unsolved problems in mathematics. It's known as Goldbach's Conjecture. It's apparently been proved to a certain point, but I don't believe a general proof has been established. You can read about it here:
http://en.wikipedia.org/wiki/Goldbach%27s_conjecture
If you DO discover a proof......you probably needn't worry about any more of your math classes......you can go straight to Harvard. P.S.......whatever you do, DON'T show the proof to your teacher.........no need to let him/her hog the credit!!!!
For the second one....I'm not sure that any "formula" exists that always produces a prime. We can produce certain primes- called Mersenne primes - by the use of an algorithm. I'm sure other "prime" algorithms exist, as well. Computer programs can be designed to "check" for the existence of a prime....but I'm out of my depth, here. Also, as I understand it, certain "found" primes are referred to as "probable" primes. But I can tell you this...that any prime ≥5 must be written in the form of either 6n+1 or 6n-1.
To see this......look at at a partial "number line"
(6n-4) (6n-3) (6n-2) (6n-1) (6n) (6n+1) (6n+2) (6n+3)(6n+4)(6n+5)(6(n+1))
Notice the 4 numbers on either side of 6n. All are divisible by either 2 or 3 except for (6n-1) or (6n+1). Now, be careful here....I'm not saying that 6n-1 or 6n+1 ARE primes. For instance, for n= 20, 6n-1 = 119 and 6n+1 =121, and neither of those are prime!! I'm just saying that if we have a prime, it can always be written in one of the two forms. Note too, that (6n-5) = (6(n-1) +1) = (6k-1). So, it might or might not be prime, as well.
For the third one.....I won't attempt to give you a number....but I'll give you something that might impress your teacher more....this was shown by a mathematician, (Euclid - maybe you've heard of him??) many years ago....
Let's assume that some of your classmates find some prime number with > 500,000,000,000 digits. (I doubt that they will...as of the moment, no prime has been found with a billion digits, much less 500 billion digits!!! It's speculated that the first billion digit prime won't be found until about 2024.) Just tell your teacher that you have something better....
Let's show that, in fact, ANY prime number we find can't possibly be all that impressive.
Again, using contradiction, let's assume that YOU have found the "greatest" prime number!! (Hey, it's GOT to be WAY larger than anything your classmates find, right??) Let's call this prime "p."
Now, let's multiply all the "known" primes togther, including the "greatest" one - "p.'
This would form a new product, > p - let's call it "m" - and let's add "1" to m = m +1.
Now, if m +1 is prime, we have some prime greater than 'p." Thus "p" ISN'T the "greatest" prime. (Sorry to disappoint you!!)
But, if m+1 ISN'T "prime," then there must be some "unknown" prime > p that we haven't included in our list of primes. Note if we divide m+1 by any of the "known" primes, we get a remainder of 1. Thus, if m+1 isn't prime, then there must be some prime > p that DOES divide it !! Thus, "p" isn't the "greatest" prime in this case, either.
Thus....there isn't any "greatest" prime, and a prime number with > 500,000,000,000 digits is just "small potatoes.''
BTW - Tell your teacher that you will complete this assignment just as soon as he/she hands you the proof of something known as the "Twin Prime Conjecture." Don't worry....your teacher get the humor!!!
