What is the smallest even number greater than three that cannot be expressed as the sum of two prime numbers?
Answers
Answer:
The smallest prime number that is the sum of two primes is 5. 5 = 2 + 3, and both 2 and 3 are primes.
Note that if a number is the sum of two primes and is also prime, one of the numbers forming the sum must be 2; because 2 is the only even prime number. All other prime numbers are odd. Hence, the sum of any two prime numbers where 2 is not one of them must be an even number greater than 2 (the sum of two odd numbers is even); hence not prime.
The question is about the smallest prime that is the sum of two primes. A related and more interesting question is about the largest? Is there the largest prime number that is the sum of two primes? This is an open question, and is the same as the question whether there are finitely many, or infinitely many “prime pairs”. A prime pair are two prime numbers with difference of 2 - such as (3, 5), (5, 7), (11, 13), (17, 19), etc. These pairs become increasingly rare as the numbers get larger. Progress has been made (by Yitang Zhang, James Maynard, Terence Tao, and others) towards proving that there are infinitely many prime pairs, which would imply there is no largest prime number that is the sum of two primes. But a complete proof has not been provided - to the best of my knowledge.
Step-by-step explanation:
It states: Every even integer greater than 2 can be expressed as the sum of two primes. ... In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that Every odd number greater than 5 can be expressed as the sum of three primes.
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