Sum of Distinct Primes: Fact or Fiction?

In summary: This one is 'harder' in the sense that it is not clear how to know when two prime numbers are distinct.
  • #1
chhitiz
221
0
i observed that every composite no. with exception of 4 and 6 can be expressed as sum of distinct prime no.s. eg: 200=103+97 100=53+47 25=13+7+5
is this true? is there any theorem stating as such?
 
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  • #2
Interesting question!

It seems true. Maybe you could prove it using induction?

The base cases could be integers from 1 to 20, say, by just showing the sums.

Then you could assume that it works for all composite numbers up to a composite number k.

Then does it work for the next composite number? Well, look at the largest prime number less than it. If composite - prime is not 1 or 4 or 6, then the sum can certainly be expressed as prime + (sum for the difference composite - prime, which can definitely be done... if the difference is prime, just take the prime, otherwise, the inductive hypothesis covers it). If the difference is 1 or 4 or 6, try the next prime number below the one you just tried. If this isn't 4 or 6, do the same thing. If it is, try the next lower prime number. And etc.

21 = composite, 19 = prime, diff = 2, done with sum = 19 + 2.

22 = composite, 19 = prime, diff = 3, done with sum = 19 + 3.

24 = composite, 23 = prime, diff = 1. Try 19 = prime, then diff = 5 and done with 19 + 5.

25 = composite, 23 = prime, diff = 2, done with 23 + 2.

26 = composite, 23 = prime, diff = 3, done with 23 + 3.

27 = composite, 23 = prime, diff = 4, try 19 = prime, diff = 9, done with 19 + psum(9) = 19 + 7 + 2.

etc.

This is just a sketch, but I think it should work. Does anybody have any comments?
 
  • #3
You can prove this with Bertrand's postulate.
 
  • #4
Yes, indeed.
It also follows from Bertrand's postulate that the number of such distinct primes does not exceed log2(N), where N is the original number to be decomposed.
Am I correct?
 
  • #5
This is almost identical to 'Goldbach's conjecture' which is more than two and a half centuries old, famous, and has never been proved, so you are unlikely to.
 
  • #6
This one is 'easier' in the sense that there is no limit on the number of primes that make the sum, just that they are distinct, whilw Goldback's conjecture restricts to two primes. It seems to me that Bertrand's postulate is sufficient for the proof.
 

1. What is the sum of distinct primes?

The sum of distinct primes is the total value obtained by adding up all the prime numbers that are unique and not repeated in a given set of numbers.

2. How do you find the sum of distinct primes?

To find the sum of distinct primes, you first need to identify all the prime numbers in the given set. Then, add up all the unique prime numbers to get the sum.

3. What is an example of finding the sum of distinct primes?

For example, if we have the set of numbers {2, 3, 5, 5, 7}, the sum of distinct primes would be 2 + 3 + 7 = 12, as 5 is repeated and is not considered as a distinct prime.

4. Why is finding the sum of distinct primes important?

Finding the sum of distinct primes is important in number theory and can help in solving various mathematical problems. It also has applications in cryptography and computer science.

5. Are there any patterns or properties of the sum of distinct primes?

There are several patterns and properties of the sum of distinct primes, such as Goldbach's conjecture, which states that every even number greater than 2 can be expressed as the sum of two distinct prime numbers. There are also various algorithms and formulas that can be used to calculate the sum of distinct primes efficiently.

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