Duo primus
What is the smallest composite number that cannot be written as the sum of 2 prime numbers ?
The answer is 27.
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2 solutions
Haha I was actually thinking of posting this problem a while back but didn't get around to it, so I'm glad to see that you've done it. Here is how the sequence of such composites continues. Of course, as Goldbach conjectures, (still an open question, but it's a good bet it's true), every integer n > 5 can be written as the sum of three primes, and indeed
2 7 = 2 3 + 2 + 2 = 1 9 + 5 + 3 = 1 7 + 7 + 3 = 1 7 + 5 + 5 =
1 3 + 1 1 + 3 = 1 3 + 7 + 7 = 1 1 + 1 1 + 5 .
Follow-up questions: Can every integer n be written as either the sum or difference of two primes? What is the cardinality of the set of integers that cannot be written as the sum of three distinct primes?
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Can u post this as a question ?
I suppose the answer to the first question is no, since 23 cannot.
However, the second one looks much more involved!
I would be happy to post it but I am not yet sure of the solution... 😐
Also, are you able to assume Goldbach's conjecture is true?
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1st question: Yes, no non-twin prime can be expressed as either the sum or difference of two primes, and 2 3 is the first of these. I suppose the better companion problem to yours is to find the first composite that cannot be expressed as either the sum or difference of two primes. The honors go to 9 3 , which lies in the middle of the first prime gap greater than 6 , (neighboring primes 8 9 and 9 7 ). I can post this one referencing your problem as inspiration, if that's o.k..
2nd question: Yeah, much tougher; tougher than Goldbach's conjecture. But looking at numbers up to 1 0 0 I find that the only ones that cannot be expressed as the sum of three distinct primes are 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 1 , 1 3 , 1 7 . (The only numbers up to 1 0 0 that cannot be written as a sum of ≥ 2 distinct primes are 1 , 2 , 3 , 4 , 6 , 1 1 , as 5 = 2 + 3 , 7 = 2 + 5 , 8 = 3 + 5 , 9 = 2 + 7 , 1 3 = 2 + 1 1 and 1 7 = 2 + 3 + 5 + 7 .) So I conjecture a cardinality of 1 2 . :)
No, while it's truth seems overwhelmingly likely, we can't assume Goldbach's conjecture to be true. There is Vinogradov's theorem, though, a weak version of Goldbach's weak theorem, which implies that any sufficiently large odd integer can be written as the sum of three primes.
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@Brian Charlesworth – I was thinking along the same lines myself... Time to write up a new paper... "The Charlesworth Conjecture" :0)
By Goldbach's Conjecture, which we know holds for all small numbers we will come across in this problem 1 , all even numbers can be written as the sum of two primes. Thus we only need to search odd, composite numbers.
Since an odd number must be the sum of an even an odd number if it is the sum of two integers, and 2 is the only even prime, we only need to check whether n − 2 is prime for each odd, composite n in ascending order:
9 − 2 = 7 , which is prime.
1 5 − 2 = 1 3 , which is prime.
2 1 − 2 = 1 9 , which is prime.
2 5 − 2 = 2 3 , which is prime.
2 7 − 2 = 2 5 , which is not prime ( 2 5 = 5 ∗ 5 ).
Thus 2 5 is the smallest composite number that cannot be written as the sum of 2 primes.
1 Note: Goldbach holds for all n < 4 × 1 0 1 8 ( Wikipedia ).
The composite numbers can be written as the sum of two primes as follows:
But, 27 can't be made with the sum of two primes. Since it is odd, one of the primes must be 2, and 27-2 = 25 which isn't prime. □