# Express numbers as the sum of two consecutive primes

Which of the following numbers can be expressed as the sum of 2 numbers consecutive primes?

I) 20 II) 36 III) 52

The correct answer is II) and III)

I solved it by explaining it as odd numbers:

36 = (2n + 1) + (2n + 3)

36 = 4n + 4 / n

8 = n

(2* 8 + 1) + (2 * 8 + 3) = 17 + 19 = 36


That worked fine, but with 52 no, since nothing assured me, that those two numbers were primes, they only assured me that they were odd.

So when doing the same with 52:

52 = 4n + 4 / n
12 = n

(2*12 + 1) + (2* 12 + 3) = 25 + 27 = 52


The sum is correct, but none is prime. The only thing I did to solve it, was to look for prime numbers around those two odd numbers. But for large numbers this will be quite tedious, then, how are these types of exercises solved regularly?

• $23+29=52$ is the second answer – Piquito Feb 24 '18 at 21:43
• I know, I'm not looking for the answer, but the way to solve it. Why if they were large numbers, I would have to look for prime numbers around them and start dividing those numbers to know if they are primes – Eduardot Feb 24 '18 at 21:50
• The difference between two consecutive primes is not bounded! – Piquito Feb 24 '18 at 21:59
• I am not sure what the point of this exercise was, but no one would dream of getting the solution for large even numbers except by computer. – almagest Feb 25 '18 at 16:58
• The starting point would be the largest odd number less than half the number of interest, but it is simply brute force after that. – herb steinberg Feb 25 '18 at 22:00

## 1 Answer

$36 = (3\cdot5+2)+(3\cdot5+4) = 17+19$

$52 = (3\cdot7+2)+(3\cdot9+2) = 23+29$