Monthly Archives: January 2018

Prime Number Problem

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Following is a problem about prime factorization of the sum of consecutive odd primes. (source: problem 80 from The Green Book of Mathematical Problems)

Prove that the sum of two consecutive odd primes is the product of at least three (possibly repeated) prime factors.

The first thing to observe is that sum of odd numbers is even, hence the sum of two consecutive odd primes will be divisible by 2. Let’s see factorization of some of the examples:

3 + 5 = 2\times 2 \times 2
5 + 7 = 2 \times 2\times 3
7+11 = 2 \times 3\times 3
11+13 = 2 \times 2 \times 2 \times 3
13+17 = 2 \times 3 \times 5
17+19 = 2\times 2\times 3 \times 3
19+23 = 42 = 2\times 3\times 7
23+29 = 52 = 2\times 2 \times 13

Now let p_n and p_{n+1} be the consecutive odd primes, then from above observations we can conjecture that either p_n+p_{n+1} is product of at least three distinct primes or p_n+p_{n+1}= 2^k p^\ell for some odd prime p such that k+\ell \geq 3.

To prove our conjecture, let’s assume that p_n+p_{n+1} is NOT a product of three (or more) distinct primes (otherwise we are done). Now we will have to show that if p_n+p_{n+1}= 2^k p^\ell for some odd prime p then k+\ell \geq 3.

If \ell = 0 then we should have k\geq 3. This is true since 3+5=8.

Now let \ell > 0. Since k\geq 1 (sum of odd numbers is even), we just need to show that k=1, \ell=1 is not possible. On the contrary, let’s assume that k=1,\ell = 1. Then p_n+p_{n+1} = 2p. By arithmetic mean property, we have

\displaystyle{p_n < \frac{p_n+p_{n+1}}{2}} = p <p_{n+1}

But, this contradicts the fact that p_n,p_{n+1} are consecutive primes. Hence completing the proof of our conjecture.


This is a nice problem where we are equating the sum of prime numbers to product of prime numbers. Please let me know the flaws in my solution (if any) in the comments.

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Riemann Hypothesis is back in news

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For details, visit Terence Tao’s blog: https://terrytao.wordpress.com/

Here is a nice discussion thread from Reddit

 

Birch and Swinnerton-Dyer Conjecture

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This is part of the 6 unsolved Millennium Problems. Following is a beautiful exposition of the statement and consequences of this conjecture:

Anybody with high-school level knowledge can benefit from this video.

What is…? and A-To-Z

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If you like reading nice expositions to mathematical terminologies, then I would suggest you to consider following two websites:

what-is

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