Please help figure this out: prime numbers largest, smallest, twin primes.

[Tru2mself]

Guys, please help me figure this out:

1) how to calculate the largest prime less than 300

2) why 35 and 37 are not twin primes?

3) the smallest number divisible by five different primes

Any input would be greatly appreciated)

 

[evinda]

Guys, please help me figure this out:

1) how to calculate the largest prime less than 300

2) why 35 and 37 are not twin primes?

3) the smallest number divisible by five different primes

Any input would be greatly appreciated)

For the first subquestion,use the sieve of Eratosthenes,to find all the primes till $300$ and so you will be able to find the largest prime,that is less than $300$.

For the second subquestion, we know that twin primes are two primes that differ from each other by two.But is it possible that $35$ and $37$ are twin primes? Note that $5 \mid 35$.

For the third subquestion,the number we are looking for is divisible by the first five different primes. Do you know which are these?
So,is the number maybe equal to the product of these primes? (Wondering)

 

[MarkFL]

I would use a slightly different approach to the first problem (although the method given above is valid). I would begin by observing:

\(\displaystyle 17^2<300<18^2\)

So, we only need to test for divisibility by the primes less than or equal to 17. So, begin with 299 and use tricks for determining divisibility:

Divisibility tricks

Then repeat for numbers you are unsure of. For example you know not to test even numbers, etc.

 

[Tru2mself]

Thanks guys, you are awesome (Nod)

I think for 1) the number is 289.

I am still not sure how to answer 2) and 3) I missed the class when we were studying primes so I am trying to figure it out myself, but I guess just not understanding the whole concept of primes (puzzled smile)

 

[evinda]

Thanks guys, you are awesome (Nod)

I think for 1) the number is 289.

I am still not sure how to answer 2) and 3) I missed the class when we were studying primes so I am trying to figure it out myself, but I guess just not understanding the whole concept of primes (puzzled smile)

No,that is not the largest prime number less than $300$. $289$ is not a prime number.Have you tried to apply the sieve of Eratosthenes? (Wondering)
Do you have any other questions about the other subquestions?

 

[Evgeny.Makarov]

I missed the class when we were studying primes so I am trying to figure it out myself, but I guess just not understanding the whole concept of primes

Sorry for being blunt, but you are asking the wrong question in post #1. Apparently, you don't know the meaning of one of the key terms in the question, and you somehow hope to answer it. Not only this is unreasonable, but this misleads people into suggesting more advanced concepts like the sieve of Eratosthenes.

I want to make clear that I am not looking down on you for not knowing something. I am saying that instead of asking for help with these problems you should be asking what prime numbers are. Of course, since this is a basic concept in mathematics, there are numerous resources on the Internet describing it, so it is probably both more reasonable and more useful to ask for links and references rather than a new explanation here. It's OK to say what kinds of resources you need, e.g., "I find Wikipedia explanation too hard and would prefer a book for beginners". It is fine to write a specific thing you don't understand in an explanation. But it's not OK to bypass learning the necessary concepts and hope that somehow the problem will become understandable.

Among Internet resources, I would recommend Wikipedia (in my opinion, it does a great job in gently introducing a concept and not just providing information for those who knew but forgot) , MathWorld, Puplemath and MathForum. Of course, there are hundreds of others.

In the end, let me quote an advice from Timothy Gowers, a Cambridge professor and a Fields Medal winner.

Let me introduce a notion of fake difficulty. Every pure maths supervisor at Cambridge has had conversations like this:

Supervisee: I found this question rather difficult.

Supervisor: Well, what were your thoughts?

Supervisee: Erm … I don’t know really, I just looked at the question and didn’t know where to start. [By the way, never say that. Ever.]

Supervisor: OK, well the question asks us to prove that the action of G on X is faithful. So what does it mean for an action to be faithful?

Supervisee: Oh … er … no, I can’t remember. Sorry.

Supervisor: Have faithful actions been defined in lectures?

Supervisee: I’m not sure. Yes, I think so.

Supervisor: But hang on, if you weren’t sure what a faithful action was, did you not think to look up the definition in your notes?

Etc. etc. This is a fake difficulty because it is not a legitimate reason to get stuck on a question. If you don’t know a definition, you can look it up. (If you can’t find it in your notes, then type it into Google and the answer will be there for you in a Wikipedia article.) “I didn’t know where to start” is a well-known euphemism for “I was too lazy even to work out what the question was asking.” If you come to a supervision with fake difficulties, then you will waste time (not just yours, but that of your supervision partner) dealing with problems that do not require external help, and you will not pick up the mathematical tips that come from engaging with real difficulties.

 

[Tru2mself]

I came to conclusion that the largest prime number that is less than 300 is 289 because 17x17 =289 Am I wrong on that?

I still am not getting what's the smallest number divisible by 5 different primes. This is kind of beyond my understanding right now, if somebody is willing to explain in tiny steps I can probably get it...(Blush)

 

[Tru2mself]

Evgeny.Makarov, no worries this is why I am here. I did exactly that - I Googled and here I am:) It gave me link to this forum and I started searching through topics to see if anybody had a similar question. I see some guys are really getting in depth on some threads. You are right I am not understanding the whole concept of prime numbers. Can you help me figure this out?

 

[MarkFL]

I agree completely with Evgeny.Makarov that Wikipedia is an excellent resource for learning. I have learned from reading their articles. Let's see what they have to say about the definition of a prime number:

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Can you see now why \(\displaystyle 17^2=289\) is not a prime number? Can you explain in your own words why 289 is not prime?