On June 7, 1742, Christian Goldbach wrote a letter to renowned mathematician Leonhard Euler. In it, he conjectured that every even number greater than 2 can be expressed as the sum of two primes. Proof of this statement, which is well known today as Goldbach’s conjecture, has evaded mathematicians to the present day, and remains one of the oldest and best known unsolved problems in mathematics.
The ternary Goldbach conjecture is a ‘weaker’ analogue to the Goldbach conjecture; it states that every odd number greater than 7 can be expressed as a sum of three primes. It is weaker in the sense that, if the original Goldbach conjecture holds, then it is automatically true. Indeed, suppose the Goldbach conjecture is true, and N is an odd number greater than 7. Then N-3 is an even number, and by the Goldbach conjecture, we can express it as a sum of two primes, N-3 = p+q. But then N = p+q+3, and we have written the odd number N as a sum of three primes.
The ternary Goldbach conjecture was proven true by Harald Helfgott in 2014. The arguments used in the proof combine ideas from all over mathematics, as is often the case in number theory. While the first author, Luke Collins, was reading for his Master’s at Warwick University he studied Helfgott’s proof under the supervision of Prof. Adam Harper, and reformulated the argument in his dissertation to outline it as concisely as possible.
The proof involves complex analysis and a clever technique called the ‘Hardy-Littlewood circle method’; the overall strategy is to show that, eventually, when our odd number N is big enough, the number of ways of writing it as a sum of three primes is at least ¼N2 (we only need this to be larger than 1 for the conjecture to be true). In fact, other attempted proofs of the conjecture throughout the 20th century applied this idea, most notably Vinogradov’s 1937 attempt. The only issue with his proof (and subsequent ones before Helfgott’s) was that ‘when N is big enough’ leaves a finite number of cases to check by hand (or by computer) until N is indeed big enough for the arguments to apply; in Vinogradov’s case, N needs to be larger than 314348907. Even though this leaves a finite number of cases to check, it is far from feasible to do so (314348907 has 6,846,169 digits). Helfgott uses a tweaked version of the circle method to improve this bound to something much more reasonable: odd numbers larger than 1027; the remaining cases can easily be checked by computer.
Luke Collins holds a B.Sc. (Hons.) degree in Mathematics and Computer Science from the University of Malta, and an M.Sc. degree in Mathematics from the University of Warwick. He will be giving a talk on the ternary Goldbach conjecture this summer for the Malta Mathematical Society. The society organises various talks accessible to the general public for all those interested in mathematics, as well as other events. Follow us on our Facebook or Instagram pages for more information.
Did you know?
• It took hundreds of years, from the 13th to the 16th centuries, for Europe to adopt the Hindu-Arabic number system. It had several advantages, two of the most prominent being that one can multiply two numbers faster than with the Roman system, and one could leave a record of the calculation. But it also had disadvantages, which are still problems today: one had to learn and remember the times tables; and one could easily make large mistakes by adding a zero to a number or by swapping two digits (in a cheque or contract, for example).
• Math is starting to be fully digitised; Kevin Buzzard’s project at Imperial College, London, called ‘Xena’, is a library of mathematics proofs created by the Microsoft computer language ‘Lean’; it is at the stage of early undergrad math.
• Several mathematicians have been imprisoned over the years. Galileo is perhaps the most famous one (house arrest), but other prominent mathematicians were Robert Recorde, Girolamo Cardano, Évariste Galois (for a day), Jean Poncelet, André Weil, and Paul Turán. On the other hand, Christopher Havens became a mathematician and published a paper while serving his 25 year prison sentence in the US.
For more trivia, see: www.um.edu.mt/think
Puzzle
• A and B play Paper-Scissors-Rock 10 times. In this game, A and B simultaneously form one of the three shapes with an outstretched hand; paper wins over rock, rock wins over scissors, and scissors wins over paper; the game is drawn if the shapes are the same. In all, A played one paper, six scissors, and three rocks, while B played the same three, four, and three times. No game was a draw. Who won?
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