Every prime greater than 7 can be written as the sum of two primes

In summary, the conversation discusses the possibility of representing a prime number greater than 7 as the sum of two primes and the subtraction of a third prime. However, it is determined that this is not always possible, as there are constraints on the values of A, B, and C. Using Goldbach's conjecture, it is possible to represent any even integer as the sum of two primes, which can then be subtracted by a prime to get the original prime number. The possibility of this method is further discussed, but no concrete proof is given.
  • #1
DbL
5
0
"Every prime greater than 7, P, can be written as the sum of two primes, A and B, and the subtraction of a third prime, C, in the form (A+B)-C, where A is not identical to B or C, B is not identical to C, and A, B, and C are less than P."

True?
 
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  • #2
Nope. Try 11.

You can't use 2 for A,B or C because the other 2 primes would be odd and you'd get an even number, so the only primes you can use are 3, 5 and 7. The largest number you can form is 7+5-3 = 9
 
  • #3
willem2 said:
Nope. Try 11.

You can't use 2 for A,B or C because the other 2 primes would be odd and you'd get an even number, so the only primes you can use are 3, 5 and 7. The largest number you can form is 7+5-3 = 9

He asked this in the homework section, and for some reason he allows the use of 1 so that 7+5-1=11 is a solution. Though, he never explains why we are allowed to use 1.

Of course, if the question is about numbers relatively prime to p, then (p-1)+2-1 is a solution to every prime. But he said that wasn't the case either.
 
  • #4
It's true for all primes between 13 and 9973.
 
  • #5
Using Goldbach's conjecture, any even integer is the sum of two primes (at least up to 1.609 × 10^18).

Meaning that (p+3) is the sum of two primes, and 3 can be subtracted to get p.
Or more generally (p+q) is the sum of two primes, where q is a prime number, and q can be subtracted to get p.

I'm not sure how you'd go about making proving it's possible when A is not equal to B.
 
  • #6
But if you subtract 3 from a prime, the result is not necessarily a prime.
 
  • #7
Right, ignore my posts, I've decided they're nonsense.
 

FAQ: Every prime greater than 7 can be written as the sum of two primes

How is it possible that every prime greater than 7 can be written as the sum of two primes?

The statement "Every prime greater than 7 can be written as the sum of two primes" is known as the Goldbach's conjecture. It has not been proven yet, but it has been tested for all primes up to 4 x 10^18 and has held true. It is widely believed to be true, but has not been formally proven yet.

Are there any exceptions to this rule?

So far, no exceptions have been found for this conjecture. However, since it has not been proven, it is still possible that there may be some rare prime numbers that cannot be written as the sum of two primes.

How can this conjecture be useful in mathematics?

This conjecture has applications in number theory and cryptography. It can also be used to generate large prime numbers by adding smaller primes together. Additionally, it has been used in computer science for testing the efficiency of algorithms for generating prime numbers.

What evidence supports the validity of this conjecture?

The Goldbach's conjecture has been tested and verified for a large number of prime numbers. Additionally, mathematicians have found results that are related to this conjecture, such as the Hardy-Littlewood asymptotic formula, which provides an approximation for the number of ways a number can be written as the sum of two primes.

How is this conjecture related to the Twin Prime Conjecture?

The Twin Prime Conjecture states that there are infinitely many pairs of prime numbers that differ by 2, such as 41 and 43. The Goldbach's conjecture is related to this because if every prime can be written as the sum of two primes, then there must be infinitely many twin primes. However, the Twin Prime Conjecture is still an unsolved problem in mathematics.

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