We also know that the integers 0 and 1, the so-called identity elements, are also perfect squares. We know that odd times odd will always generate odd, and we also know that the sequence of odds sum progressively to the sequence of perfect squares. When you realize that fact, then you may make the conjecture that every perfect square integer can be expressed as the sum of an odd semiprime and another perfect square. The Goldbach pairs (Goldbach partitions), when considered multiplicatively rather than additively, can be used to generate what are known as odd semiprimes - the products of two odd prime factors (unique or identical). See here to read about some more, and here to find out more about the Goldbach conjecture and our Goldbach calculator. The weak and strong Goldbach conjectures are just two of many questions from number theory that are easy to state but very hard to solve. Until very recently the result had only been verified for odd numbers greater than 2 x 10 1346 - that's a number with 1,347 digits! But then, in 2013, the Peruvian mathematician Harald Helfgott closed the enormous gap and proved that the result is true for all odd numbers greater than 5. Again we can see that this is true for the first few odd numbers greater than 5: The weak Goldbach conjecture says that every odd whole number greater than 5 can be written as the sum of three primes. There is a similar question, however, that has been proven. The latest result, established using a computer search, shows it is true for even numbers up to and including 4,000,000,000,000,000,000 - that's a huge number, but for mathematicians it isn't good enough. Goldbach conjecture is true for even numbers up to and including 100,000. That wasn't very wise of Euler: the Goldbach conjecture, as it's become known, remains unproven to this day. He wrote about his idea to the famous mathematician Leonhard Euler, who at first treated the letter with some disdain, regarding the result as trivial. This is also the conclusion that the Prussian amateur mathematician and historian Christian Goldbach arrived at in 1742. If you keep on trying you will find that it seems that every even number greater than 2 can indeed be written as the sum of two primes.
8 = 3 + 5, 5 is a prime too, so it's another "yes".6 = 3 + 3 and 3 is prime, so it's "yes" for 6 also.4 = 2 + 2 and 2 is a prime, so the answer to the question is "yes" for the number 4.Here is one of the trickiest unanswered questions in mathematics:Ĭan every even whole number greater than 2 be written as the sum of two primes?Ī prime is a whole number which is only divisible by 1 and itself.
Leonard Euler (1707-1783) corresponded with Christian Goldbach about the conjecture now named after the latter.
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