The discussion revolves around identifying prime numbers of the form \(2^n + 1\). Participants explore examples and the implications of these primes, particularly in relation to Mersenne primes.
There is an ongoing exploration of how to articulate the findings related to the primes \(17\) and \(257\). Some participants suggest that rigorous proof may not be necessary, while others provide insights into how to approach the proof of primality.
Participants note that the problem may not require extensive proof beyond demonstrating that the examples found are prime. There is also mention of the expectations from the grader regarding the level of detail required in the explanation.
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2n − 1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2p − 1 for some prime p.
So do I just show my work and answer for this problem without proof?Office_Shredder said:Are you asking how to explain how you found these, or how to prove they are prime numbers.
You probably don't need to explain how you came up with them, or you can just write "I tried some examples and here's what I found" if you're worried about a particularly harsh grader.
Math100 said:So do I just show my work and answer for this problem without proof?
## 17=2^{4}+1 ##
## 257=2^{8}+1 ##
Thank you!Office_Shredder said:I don't know what your grader is expecting, but my guess is this is fine. If you think you need to be very rigorous on proving the primeness, it's not that hard. I'll do 17 for you so you can get the feel for it.
Here's a useful lemma: if ##n## is a composite number, then there exists a prime ##p## such that ##p|n## and ##p^2 \leq n##
Proof: factor ##n=mk##. One of ##m## or ##k## is smaller (or they are equal) wlog let it be ##k\leq m##. Then ##k^2\leq km=n##. And there must be some prime ##p## that divides ##k##, and hence divides ##n##. If ##k=pq##, then ##p^2 \leq k^2 \leq n##.
Since ##5^2 > 17##, if 17 is composite it must be divisible by a prime less than or equal to 4. 1 and 4 are not primes, so we only need to check 2 and 3. But ##17= 2*8+1## and ##17=3*5+2## demonstrates 17 is not divisible by either.
Again though, I would be very surprised if you were supposed to actually prove this, though I don't know the level of dumb details your course requires you to provide. I would guess just what you wrote is sufficient. (What I wrote is assuming a bunch of properties about how multiplication preserves ordering, so if you're a real stickler you could also prove things like ##k\leq m \rightarrow k^2 \leq km##
All that the problem is asking you to do is to find two more primes of the given form. There is nothing that you need to prove, other than has already been stated about showing that your found examples are indeed prime.Math100 said:Find two more of these primes.
Now I'm stuck.
Just say primes of the form ##2^k+1##. The only purpose this convoluted phrasing serves is confusing or annoying the reader.Math100 said:Homework Statement:: An unanswered question is whether there are infinitely many primes that are ## 1 ## more than a power of ## 2 ##, such as ## 5=2^2+1 ##. Find two more of these primes.
Every composite number has a prime factor less than or equal to its square root. To check ##257## is prime you need only divide by ##3, 7, 11## and ##13##. You can see by inspection it's not divisible by ##2## or ##5##.Math100 said:So do I just show my work and answer for this problem without proof?
## 17=2^{4}+1 ##
## 257=2^{8}+1 ##