Prime time!
What is the smallest non-negative integer that can't be expressed as the sum or difference of two prime numbers?
If you think there is no such number, please provide the answer as 99999.
The answer is 23.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
By Brun's Theorem almost all primes are such "isolated" primes, (i.e., not part of a twin prime pair).
(Also, by Goldbach's conjecture, (though not proved it is likely valid), we can eliminate all the evens right off the bat. :))
Log in to reply
Ah, Goldbach's conjecture... I didn't realize that all even numbers could be written as the sum of two primes, I wonder about the difference of two primes...
Log in to reply
Well, Polignac's conjecture remains unproven, but that would be a stronger result than we need here, as we just at least one prime gap of length 2 n and we aren't restricting ourselves to consecutive primes. I believe Chen Jingrun at some point proved that every even integer can be expressed as the difference of two (not necessarily consecutive) primes but a citation eludes me at the moment.
One can really go down a rabbit-hole while pursuing open questions regarding primes. It's fun to do every once in a while, but you have to guard against going in too deep. :O
It is pretty trivial to find solutions for all the non-negative integers up through 2 2 .
I agree that it's trivial, but I think it would help if you wrote up those list of integers to convincingly prove that 23 is indeed the answer.
Log in to reply
Done. Hows that?
Log in to reply
Potential follow-up question: What is the smallest non-negative composite integer that cannot be expressed as the sum or difference of primes?
For this we need to find the first pair of successive primes ( p , q ) such that q − p > 6 . Then p + 4 will be the smallest such composite. The pair ( 8 9 , 9 7 ) is the first to fit the bill, giving us 9 3 as the answer.
Yupppers! =D
Why the solution cannot be 0?
Log in to reply
The primes involved do not have to be distinct, so we can have p − p = 0 for any prime p .
Sorry, found the answer
For 1 to 22 you have:
For 2 3 , no sum or difference can be found, since the sum or difference would have to contain an even number, and the only even prime is two. However, neither 2 1 nor 2 5 is prime.
Therefore, 2 3 is the smallest such integer.