4 distinct prime numbers
What is the smallest positive integer N that satisfies N = a + b = c × d , where a , b , c , and d are distinct prime numbers?
The answer is 10.
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1 solution
Hmmm... Conjecture: Every integer of the form 2 p for prime p ≥ 5 can be written as the sum of two distinct primes. Weaker than Goldbach but still not proven, as far as I know.
(Edited in light of Calvin Lin's comment below.)
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I would certainly bet money on it... I think its time to write up a paper proving some of these conjectures once and for all! :0)
Do you mean semiprime (product of 2 primes), or do you mean numbers of the form 2 p ?
E.g. 5 × 7 is not the sum of 2 primes
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Quite right. I started out with the 2 p conjecture and then casually extended it to semiprimes without giving it enough thought. I'll edit my comment accordingly.
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@Brian Charlesworth – As an aside, Goldbach's weak conjecture is that every odd number greater than 5 can be expressed as the sum of 3 primes.
This was proved in 2013.
How do you know that the answer cannot be 1,2,3,4,5,6,7,8 or 9?
Maybe you'd better reduce the number of letters.
Likely, how about using c d instead of c × d ?
Most people do not write × sign.(I dont know the reason.)
1 0 = 3 + 7 = 2 × 5
This is the smallest such number, since the number needs to be greater than all four primes, and the first four primes are 2, 3, 5, and 7. So it would need to be greater than 7. Also, the number needs to be the product of two distinct primes. Therefore 8 and 9 or out.