[Math] Prove: There exists an even integer $n$ that can be written in two different ways as a sum of two distinct primes.

discrete mathematicselementary-number-theory

I am working on this problem,

Prove: There exists an even integer $n$ that can be written in two different ways
as a sum of two distinct primes.

I know:

$3+13=11+5=16$

$11+13=7+17=24$

$23+7=11+19=30$

I don't see any information to help me do the question from my examples, can anyone give me a hint or suggestion?

Thanks in advance!

Best Answer

You had already done what was required. The question asks you to prove that there exists an even integer which can be written in two different ways as the sum of two distinct primes. So, your mission is to find such one. Any of your examples would suffice as an answer.

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[Math] Can every even integer greater than four be written as a sum of two twin primes

In fact, it was already a conjecture; mathworld says, "It is conjectured that every even number is a sum of a pair of twin primes except a finite number of exceptions whose first few terms are $2, 4, 94, 96, 98, 400, 402, 404, 514, 516, 518,\cdots$" ... (OEIS A007534; Wells 1986, p. 132).

[Math] Prove that every integer $n\geq 7$ can be expressed as a sum of distinct primes.

We shall inductively prove a stronger form, namely that every positive integer $n \ge 7$ can be written as the sum of distinct primes such that the largest is at most $\max(11,n-7)$. It turns out that strengthening makes the induction work!

Take $n \ge 28$.

Let $m = \lceil \frac{n-6}{2} \rceil= \lfloor\frac{n-5}{2} \rfloor$.

Let $p$ be a prime such that $m+1 \le p \le 2m-1$ [by Bertrand's postulate].

Then $\frac{n-5}{2} \le p \le n-7$.

By the induction to be established $n-p$ can be written as a sum of distinct primes such that the largest is at most $\max(11,n-p-7)$, which is less than $p$ because $p \ge \frac{28-5}{2} > 11$ and $2p \ge n-5 > n-7$.

Thus $n$ can be written as a sum of distinct primes such that the largest is at most $\max(7,n-7)$, and the induction holds as long as the claim is true for every $n$ from $7$ to $27$.

7 = 5+2
8 = 5+3
9 = 7+2
10 = 5+3+2
11 = 11
12 = 7+5
13 = 11+2
14 = 7+5+2
15 = 7+5+3
16 = 11+5
17 = 7+5+3+2
18 = 11+7
19 = 11+5+3
20 = 11+7+2
21 = 11+7+3
22 = 13+7+2
23 = 13+7+3
24 = 13+11
25 = 13+7+5
26 = 13+11+2
27 = 13+11+3

Best Answer

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[Math] Can every even integer greater than four be written as a sum of two twin primes

[Math] Prove that every integer $n\geq 7$ can be expressed as a sum of distinct primes.

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