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Number Properties (Advanced) - Sum of Prime Numbers

The discussion that follows doesn’t fall in the purview of GMAT and you needn’t know it. You will be able to solve any question without taking this post into account but that has never stopped us from letting loose our curiosity so here goes…


Question 1: Which of the following CANNOT be the sum of two prime numbers?

(A) 19
(B) 45
(C) 58
(D) 79
(E) 88


Solution: We discussed in that post that the sum of two prime numbers is usually even because prime numbers are usually odd. We also discussed that if the sum of two prime numbers is odd, it means one of the prime numbers is certainly 2 – the only even prime number.


For example:

2 + 3 = 5

2 + 7 = 9

2 + 17 = 19


Then it makes perfect sense to first look at the options which are odd. To be sum of two prime numbers, the sum must be of the form 2 + Another Prime Number.

We saw that (D) 79 = 2 + 77 (77 is not prime.) and hence we got (D) as our answer.


Now the question we raised there was: What happens if instead of 79, we had 81?


81 = 2 + 79


Then all three odd options would have been sum of two prime numbers and we would have needed to check the even options too. How do you figure whether an even number can be written as the sum of two prime numbers?


This is where Goldbach’s Conjecture comes into play (you don’t really need to know it. We are doing it for intellectual purposes. GMAC will never put you in this fix).


It says “Every even integer greater than 2 can be expressed as the sum of two primes.


Mind you, it’s a conjecture i.e. it hasn’t been proven for all even numbers (only for even numbers till 4 * 10^{18}) but it does seem to hold.

For example:

4 = 2 + 2

6 = 3 + 3

8 = 3 + 5

10 = 3 + 7 = 5 + 5

12 = 5 + 7

and so on…


So given any even sum greater than 2, you can say that it CAN be written as sum of two prime numbers, for all practical purposes.


In fact, and here we are going into really geeky territory, we expect that every large even integer has not just one representation as the sum of two primes, but in fact has very many such representations. For all we know, 6 may be the only even number greater than 2 which cannot be written as the sum of two distinct prime numbers.


Coming back to our original question, we will actually check only odd numbers to see whether they can be written as sum of two primes. One of them has to be such that it cannot be written as sum of two primes and finding that is very simple! 


So all in all, the question that seemed very tedious turned out to be very simple!


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