Prime Number Problem

Standard

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.

9 responses

  1. I ‘discovered’ inadvertently that odd prime numbers are the only numbers that are the sum of two and only two consecutive integers. Do you happen to know how this description is defined in formal mathematical language?

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  2. I went through it in detail. It seems quite an elegant solution. Similarly, you should also try the Scottish Book of problems (collections of problems posed and solved by Polish mathematicians like Stan Ulam, Stefan Banach…etc.) when Poland was a separate country before world war II. There is one more book “Red book of problems”…hopefully, all these books are there in your library…Once again, a very good attempt…:-)

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