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Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics.It states that every even natural number greater than 2 is the sum of two prime numbers.
This conjecture is called "weak" because if Goldbach's strong conjecture (concerning sums of two primes) is proven, then this would also be true. For if every even number greater than 4 is the sum of two odd primes, adding 3 to each even number greater than 4 will produce the odd numbers greater than 7 (and 7 itself is equal to 2+2+3).
Goldbach's weak conjecture, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of Goldbach's conjecture. Ivan Vinogradov proved it for large enough n (Vinogradov's theorem) in 1937, [1] and Harald Helfgott extended this to a full proof of Goldbach's weak conjecture in 2013. [2] [3] [4]
Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes.” You check this in your head for small numbers: 18 is 13+5, and 42 is 23+19.
Schnirelmann sought to prove Goldbach's conjecture. In 1930, using the Brun sieve, he proved that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant. [1] [2] His other fundamental work is joint with Lazar Lyusternik.
But with Goldbach's conjecture, along with the fact that P would immediately know X and Y if their product were a semiprime, it can be deduced that the sum x+y cannot be even, since every even number can be written as the sum of two prime numbers. The product of those two numbers would then be a semiprime.
His work on the twin prime conjecture, Waring's problem, Goldbach's conjecture and Legendre's conjecture led to progress in analytic number theory. In a 1966 paper he proved what is now called Chen's theorem : every sufficiently large even number can be written as the sum of a prime and a semiprime (the product of two primes) – e.g., 100 = 23 ...
In mathematics, the Goldbach–Euler theorem (also known as Goldbach's theorem), states that the sum of 1/(p − 1) over the set of perfect powers p, excluding 1 and omitting repetitions, converges to 1: