Numberphile - Videos about Numbers and Stuff. Prime Conjectures and Open Question. Nombres premiers - table jusqu'à 1000. In Their Prime: Mathematicians Come Closer to Solving Goldbach's Weak Conjecture. One of the oldest unsolved problems in mathematics is also among the easiest to grasp.
The weak Goldbach conjecture says that you can break up any odd number into the sum of, at most, three prime numbers (numbers that cannot be evenly divided by any other number except themselves or 1). For example: Every odd integer larger than 1 is the sum of at most five primes. I’ve just uploaded to the arXiv my paper “Every odd number greater than 1 is the sum of at most five primes“, submitted to Mathematics of Computation.
The main result of the paper is as stated in the title, and is in the spirit of (though significantly weaker than) the even Goldbach conjecture (every even natural number is the sum of at most two primes) and odd Goldbach conjecture (every odd natural number greater than 1 is the sum of at most three primes). It also improves on a result of Ramaré that every even natural number is the sum of at most six primes. This result had previously also been established by Kaniecki under the additional assumption of the Riemann hypothesis, so one can view the main result here as an unconditional version of Kaniecki’s result.
The method used is the Hardy-Littlewood circle method, which was for instance also used to prove Vinogradov’s theorem that every sufficiently large odd number is the sum of three primes. The Prime Pages (prime number research, records and resources) Nombres, curiosités, théorie et usages: page d'orientation générale. Les Nombres : Histoire et évolution.