Home » Ask & Discuss » Mathematics. » Algebra « Back to Discussion

Algebra

Blazing goIITian

Joined:	21 Mar 2009
Post:	345

30 Apr 2009 04:40:55 IST

0 People liked this

335

really good questions - a collection ( i will post all questions here )

None

1]In 1772 Euler discovered the curious fact that $n^2 +n+41$ is prime when $n$ is any of $0,1,2, \cdots, 39$ . Show that there exist $40$ consecutive integer values of $n$ for which this polynomial is not prime.

Share this article on:

Comments (2)

apollo

Blazing goIITian

Joined: 21 Mar 2009

Posts: 345

30 Apr 2009 04:41:18 IST

Like 0 people liked this

2]Show that there are infinitely many primes.

3]Find all natural numbers $n$ for which every natural number whose decimal representation has $n-1$ digits $1$ and one digit $7$ is prime.

4]Prove that there do not exist polynomials $P$ and $Q$ such that
$\pi(x)=\frac{P(x)}{Q(x)}$
for all $x\in\mathbb{N}$ .

Anant Kumar

Forum Expert

Joined: 10 Jul 2008

Posts: 598

30 Apr 2009 07:05:45 IST

Like 1 people liked this

2) Its straight forward. Assume that there is some greatest prime. Call it p_n. Let p₁, p₂, . . ., p_n-1 be the prime lesser than p_n.

The product p₁p₂p₃....p_n + 1 is the then obviously a prime and greater than p_n, which is a contradiction. Hence, there are infinitely many prime.

Quick Reply

Reply

	Some HTML allowed.
	Keep your comments above the belt or risk having them deleted.
	Signup for a avatar to have your pictures show up by your comment
	If Members see a thread that violates the Posting Rules, bring it to the attention of the Moderator Team