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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). In 2013, Harald Helfgott released a proof of Goldbach's weak conjecture. [2]
For instance, if m is odd, then n − m is also odd, and if m is even, then n − m is even, a non-trivial relation because, besides the number 2, only odd numbers can be prime. Similarly, if n is divisible by 3, and m was already a prime other than 3, then n − m would also be coprime to 3 and thus be slightly more likely to be prime than a ...
Depending on the sources used for the list of interesting numbers, a variety of other numbers can be characterized as uninteresting in the same way. [7] For instance, the mathematician and philosopher Alex Bellos suggested in 2014 that a candidate for the lowest uninteresting number would be 224 because it was, at the time, "the lowest number ...
2013 Ternary Goldbach conjecture: Every odd number greater than 5 can be expressed as the sum of three primes. 2014 Proof of Erdős discrepancy conjecture for the particular case C=2: every ±1-sequence of the length 1161 has a discrepancy at least 3; the original proof, generated by a SAT solver, had a size of 13 gigabytes and was later ...
Every positive integer has a unique factorization into a square-free number r and a square number s 2. For example, 75,600 = 2 4 3 3 5 2 7 1 = 21 ⋅ 60 2 . Let N be a positive integer, and let k be the number of primes less than or equal to N .
The classical proof. It is sufficient to prove the theorem for every odd prime number p. This immediately follows from Euler's four-square identity (and from the fact that the theorem is true for the numbers 1 and 2). The residues of a 2 modulo p are distinct for every a between 0 and (p − 1)/2 (inclusive). To see this, take some a and define ...
The full statement of Vinogradov's theorem gives asymptotic bounds on the number of representations of an odd integer as a sum of three primes. The notion of "sufficiently large" was ill-defined in Vinogradov's original work, but in 2002 it was shown that 10 1346 is sufficiently large.
They showed that every CN group of odd order is solvable. Their proof is similar to Suzuki's proof. It was about 17 pages long, which at the time was thought to be very long for a proof in group theory. The Feit–Thompson theorem can be thought of as the next step in this process: they show that there is no non-cyclic simple group of odd order ...