MHB Is there an other way to show the proposition ?

  • Thread starter Thread starter evinda
  • Start date Start date
AI Thread Summary
The discussion revolves around proving the proposition that the product of all primes \( p \) less than or equal to \( 2n \) is greater than \( 2^n \) for all \( n \geq 2 \). Participants explore whether checking specific values of \( n < 100 \) constitutes a formal proof and confirm that all necessary values have been verified. They also discuss the implications of prime distributions between certain ranges and how these affect the product calculations. The conversation emphasizes the need for thorough verification across the specified range to establish the proposition's validity. Ultimately, the participants agree on the correctness of the reasoning presented in their calculations.
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
Hello! (Wave)

I am looking at the proof that $\prod_{p \leq 2n}p>2^n, \forall n \geq 2$, where the product extends over all primes $p \leq 2n$.

For $n<100$ the above proposition is shown as follows.

View attachment 7678

I was wondering if there is also an other way to prove the proposition for $n<100$.

Also is it a formal proof to just check that the proposition is true for some values $n<100$, as it is done at the picture above? (Thinking)
 

Attachments

  • inspection.JPG
    inspection.JPG
    39 KB · Views: 108
Mathematics news on Phys.org
evinda said:
I was wondering if there is also an other way to prove the proposition for $n<100$.

Hey evinda! (Smile)

I'm afraid that I wouldn't know.
If we could, perhaps we could prove it for all $n$ in general! (Happy)

evinda said:
Also is it a formal proof to just check that the proposition is true for some values $n<100$, as it is done at the picture above? (Thinking)

For a formal proof up to $n<100$, we need to verify it for all those values of $n$.
So let's take a look if we missed one.
We seem to be missing $n=4$, so let's see...
For $n=3$ we have that $\Pi p > 2^4$, so I think that for $n=4$ we will also have that $\Pi p > 2^4$ won't we?
So then we didn't miss $n=4$ after all. (Thinking)
 
I like Serena said:
For a formal proof up to $n<100$, we need to verify it for all those values of $n$.
So let's take a look if we missed one.
We seem to be missing $n=4$, so let's see...
For $n=3$ we have that $\Pi p > 2^4$, so I think that for $n=4$ we will also have that $\Pi p > 2^4$ won't we?
So then we didn't miss $n=4$ after all. (Thinking)

I see...
Then, is it right as follows?

For $n=8$ we have that $\Pi p > 2^{14}$, so for $n=9, \dots, 14$ we will also have that $\Pi p > 2^{14}$ .

For $n=15$ we have that $\Pi p > 2^{30}$, so for $n=16, \dots, 30$ we will also have that $\Pi p > 2^{30}$ .

For $n=30$ we have that $\Pi p > 2^{65}$, so for $n=31, \dots, 65$ we will also have that $\Pi p > 2^{65}$ .

For $n=65$ we have that $\Pi p > 2^{149}$, so for $n=66, \dots, 99$ we will also have that $\Pi p > 2^{149}$ .For example for $n=99$ we want to have $\Pi p>2^{90}$ which is true since it holds that $\Pi p > 2^{149}$.

Am I right? (Thinking)
 
evinda said:
I see...
Then, is it right as follows?

For $n=8$ we have that $\Pi p > 2^{14}$, so for $n=9, \dots, 14$ we will also have that $\Pi p > 2^{14}$ .

For $n=15$ we have that $\Pi p > 2^{30}$, so for $n=16, \dots, 30$ we will also have that $\Pi p > 2^{30}$ .

For $n=30$ we have that $\Pi p > 2^{65}$, so for $n=31, \dots, 65$ we will also have that $\Pi p > 2^{65}$ .

For $n=65$ we have that $\Pi p > 2^{149}$, so for $n=66, \dots, 99$ we will also have that $\Pi p > 2^{149}$ .For example for $n=99$ we want to have $\Pi p>2^{90}$ which is true since it holds that $\Pi p > 2^{149}$.

Am I right? (Thinking)

Yep. (Nod)

It seems that whoever wrote the proof, continued each time where the condition might be violated, but wasn't violated after all. (Nerd)
 
I like Serena said:
Yep. (Nod)

It seems that whoever wrote the proof, continued each time where the condition might be violated, but wasn't violated after all. (Nerd)

Yes, for example after $n=15$ we could check the value $n=31$ instead of $n=30$, right? (Thinking)

Also, do you maybe have an idea what is meant with the following?

For establishing the table we have used the fact that between $30$ and $60$ there are $7$ prime numbers ( and that $31 \cdot 37>2^{10}$), between $60$ and $130$ there are $14$.
How do we use these facts? (Thinking)
 
evinda said:
Yes, for example after $n=15$ we could check the value $n=31$ instead of $n=30$, right?

Yep. (Nod)

evinda said:
Also, do you maybe have an idea what is meant with the following?

For establishing the table we have used the fact that between $30$ and $60$ there are $7$ prime numbers ( and that $31 \cdot 37>2^{10}$), between $60$ and $130$ there are $14$.
How do we use these facts?

Don't we have
$$\displaystyle\prod_{p<2\cdot 15} p > 2^{30}$$
so that
$$\displaystyle\prod_{p<2\cdot 30} p \quad=\quad \prod_{p<2\cdot 15} p \cdot \prod_{2\cdot 15 \le p<2\cdot 30} p\quad>\quad 2^{30} \cdot \prod_{2\cdot 15 \le p<2\cdot 30} p \quad=\quad 2^{30} \cdot 31 \cdot ... \cdot 59$$
(Wondering)
 
I like Serena said:
Yep. (Nod)
Don't we have
$$\displaystyle\prod_{p<2\cdot 15} p > 2^{30}$$
so that
$$\displaystyle\prod_{p<2\cdot 30} p \quad=\quad \prod_{p<2\cdot 15} p \cdot \prod_{2\cdot 15 \le p<2\cdot 30} p\quad>\quad 2^{30} \cdot \prod_{2\cdot 15 \le p<2\cdot 30} p \quad=\quad 2^{30} \cdot 31 \cdot ... \cdot 59$$
(Wondering)

I see... but do we need this for establishing the table?
Doesn't $\displaystyle\prod_{p<2\cdot 15} p > 2^{30}$ directly imply that $\displaystyle\prod_{p<2\cdot 30} p \geq \displaystyle\prod_{p<2\cdot 15} p >2^{30}$ ? (Thinking)
 
evinda said:
I see... but do we need this for establishing the table?
Doesn't $\displaystyle\prod_{p<2\cdot 15} p > 2^{30}$ directly imply that $\displaystyle\prod_{p<2\cdot 30} p \geq \displaystyle\prod_{p<2\cdot 15} p >2^{30}$ ? (Thinking)

Indeed, for $n=30$ we do not need it.
But for $n=31$ up to $n=60$ we do, don't we? (Wondering)
And it actually works up to $n=65$, since the table lists $2^{65}$ for it.
 
I like Serena said:
But for $n=31$ up to $n=60$ we do, don't we? (Wondering)
And it actually works up to $n=65$, since the table lists $2^{65}$ for it.

Why? I haven't understood it... (Worried)I thought that it would be as follows.

$\prod_{p \leq 2 \cdot 30} p=2^{65}>2^{64}$ and so $\prod_{p \leq 2 \cdot \lambda} p>2^{64}, \forall \lambda \in \{ 31,\dots, 64\}$. Am I wrong? (Thinking)
 
  • #10
evinda said:
Why? I haven't understood it... (Worried)I thought that it would be as follows.

$\prod_{p \leq 2 \cdot 30} p=2^{65}>2^{64}$ and so $\prod_{p \leq 2 \cdot \lambda} p>2^{64}, \forall \lambda \in \{ 31,\dots, 64\}$. Am I wrong? (Thinking)

Huh? :confused:
It seems to me that you did understand it!
It's only that instead of $\prod_{p \leq 2 \cdot 30} p=2^{65}$ we have:
$$\displaystyle\prod_{p\le 2\cdot 30} p \quad=\quad \prod_{p\le 2\cdot 15} p \cdot \prod_{2\cdot 15 < p \le 2\cdot 30} p\quad>\quad 2^{30} \cdot \prod_{2\cdot 15 < p\le 2\cdot 30} p \quad=\quad 2^{30} \cdot 31 \cdot ... \cdot 59 \quad>\quad 2^{65}$$
So:
$$\displaystyle\prod_{p\le 2\cdot 30} p > 2^{65}$$
Can you clarify? (Wondering)
 
  • #11
I like Serena said:
Huh? :confused:
It seems to me that you did understand it!

So is that what I wrote in post #9 correct? (Thinking)

I like Serena said:
It's only that instead of $\prod_{p \leq 2 \cdot 30} p=2^{65}$ we have:
$$\displaystyle\prod_{p\le 2\cdot 30} p \quad=\quad \prod_{p\le 2\cdot 15} p \cdot \prod_{2\cdot 15 < p \le 2\cdot 30} p\quad>\quad 2^{30} \cdot \prod_{2\cdot 15 < p\le 2\cdot 30} p \quad=\quad 2^{30} \cdot 31 \cdot ... \cdot 59 \quad>\quad 2^{65}$$
So:
$$\displaystyle\prod_{p\le 2\cdot 30} p > 2^{65}$$

I see... (Nod)
 
  • #12
evinda said:
So is that what I wrote in post #9 correct?

Yep. (Nod)
 
Back
Top