4 distinct prime numbers

What is the smallest positive integer N N that satisfies N = a + b = c × d , N = a+b = c \times d, where a , b , c , a,b,c, and d d are distinct prime numbers?


The answer is 10.

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1 solution

Geoff Pilling
Mar 3, 2017

10 = 3 + 7 = 2 × 5 \boxed{10} = 3 + 7 = 2 \times 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.

Hmmm... Conjecture: Every integer of the form 2 p 2p for prime p 5 p \ge 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.)

Brian Charlesworth - 4 years, 3 months ago

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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)

Geoff Pilling - 4 years, 3 months ago

Do you mean semiprime (product of 2 primes), or do you mean numbers of the form 2 p 2p ?

E.g. 5 × 7 5 \times 7 is not the sum of 2 primes

Calvin Lin Staff - 4 years, 3 months ago

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Quite right. I started out with the 2 p 2p conjecture and then casually extended it to semiprimes without giving it enough thought. I'll edit my comment accordingly.

Brian Charlesworth - 4 years, 3 months ago

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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.

Calvin Lin Staff - 4 years, 3 months ago

How do you know that the answer cannot be 1,2,3,4,5,6,7,8 or 9?

Pi Han Goh - 4 years, 3 months ago

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Thanks... I've added this to the solution.

Geoff Pilling - 4 years, 3 months ago

Maybe you'd better reduce the number of letters.

Likely, how about using c d cd instead of c × d c \times d ?

Most people do not write × \times sign.(I dont know the reason.)

. . - 2 months, 1 week ago

And

15 = 2 + 13 = 3 × 5 15 = 2 + 13 = 3 \times 5 can be also answer. :)

. . - 2 months, 1 week ago

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