Need help for the proofs about number theory.

Note: $p_{n}$ is the n-th prime number ( $p_{1} = 2, p_{2} = 3, p_{3} = 5$ , etc).

1.) Prove that $p_{n+1} < \sum\limits_{i=1}^{n} p_{i}$ for any integer $n > 2$ .

2.) Prove that $\sum\limits_{i=1}^{n} p_{i} < \prod\limits_{i=1}^{n} p_{i}$ for any integer $n > 1$ .

3.) Prove or counterexample that $\sqrt{n!} \notin \mathbb{N}$ for any integer $n > 1$ .

#NumberTheory #Bertrand'sPostulate #PrimeNumbers

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I just proved number 3 in 3 seconds lol. Just Bertrand's Postulate that there exists a prime number from $\frac{n}{2}$ to $n$ if $n$ is even or from $\frac{n-1}{2}$ to $n-1$ if $n$ is odd.

For even number $n$ , if $p > \frac{n}{2}$ , then $2p > n$ . Which means the next number that has a factor $p$ doesn't exist in $n,n-1,n-2,...,3,2,1$ any more. So $n!$ only have 1 factor of $p$ and makes it not perfect square.

For odd number $n$ , if $p > \frac{n-1}{2}$ , then $2p > n-1$ . Same as above, there exists only 1 factor $p$ .

How about $n$ ? Can $n$ have a factor of $p$ ? No. If $2p = n$ , then $n$ is an odd number, which contradicts the given. So $2p > n$ or $2p < n$ .

For $2p > n$ , we're done. $2p$ doesn't exist in $n,n-1,n-2,...,3,2,1$ any more.

For $2p < n$ , since $2p > n-1$ , therefore $2p$ can't be integers, which contradicts the truth.

Therefore, $\sqrt{n!}$ can't be integers for any $n > 1$ .

Samuraiwarm Tsunayoshi - 7 years, 3 months ago

I agree with your q3 solution, it is very common in questions like these to use Bertrand's Postulate. In a moment I will post a very nice problem indeed. q2 is very susceptible to induction (just simple induction, maybe with a contradiction element if you want to simplify things). q1 is a very nice problem, as it combines techniques used in both q2 and q3 (use induction, with a contradiction, then use Bertrand's Postulate), so I thank you for these series of problems. If you want to know the problem I previously mentioned, just visit my page.

TheKnee OfJustice - 7 years, 3 months ago

You can have a one-liner inductive proof of Problem 1) using Bertrand's postulate, which is a well known proven result in number theory.

The base case for $n=3$ is easy. Assume the assertion to hold good for an integer $n\geq 3$ . Using this postulate, we have $p_{n+1} < p_{n+2} < 2 p_{n+1}= p_{n+1}+ p_{n+1} < \sum_{i=1}^{n+1}p_i$ , where the last inequality follows from the induction assumption.

Abhishek Sinha - 7 years, 3 months ago

Bertrand's Postulate

Rajeh Alghamdi - 7 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$