most three primes, it is not always possible to find such a sum using a greedy algorithm that uses the largest possible prime at each step. The Pillai sequence tracks the numbers requiring the largest number of primes in their greedy representations. Similar problems to Goldbach's conjecture exist in which primes are replaced by other particular sets of numbers, such as the squares: It was proven by Lagrange that every positive integer is the sum of four squares. See Waring's problem and the related Waring–Goldbach problem on sums of powers of primes. Hardy and Littlewood listed as their Conjecture I: "Every large odd number (n > 5) is the sum of a prime and the double of a prime" (Mathematics Magazine, 66.1 (1993): 45–47). This conjecture is known as Lemoine's conjecture and is also called Levy's conjecture. The Goldbach conjecture for practical numbers, a prime-like sequence of integers, was stated by Margenstern in 1984, and proved by Melfi in 1996: every even number is a sum of two practical numbers. A strengthening of the Goldbach conjecture proposed by Harvey Dubner states that every even integer greater than 4,208 is the sum of two twin primes. Only 34 even integers less than 4,208 are not the sum of two twin primes. Dubner has verified computationally that this list is complete up to . A proof of this stronger conjecture would not only imply Goldbach's conjecture, but also the twin prime conjecture. In popular culture Goldbach's Conjecture () is the title of the biography of Chinese mathematician and number theorist Chen Jingrun, written by Xu Chi. The conjecture is a central point in the plot of the 1992 novel Uncle Petros and Goldbach's Conjecture by Greek author Apostolos Doxiadis, in the short story "Sixty Million Trillion Combinations" by Isaac Asimov and also in the 2008 mystery novel No One You Know by Michelle Richmond. Goldbach's conjecture is part of the plot of the 2007 Spanish film Fermat's Room. References Further reading Terence Tao proved that all odd numbers are at most the sum of five primes. Goldbach Conjecture at MathWorld. External links Goldbach's original letter to Euler — PDF format (in German and Latin) Goldbach's conjecture, part of Chris Caldwell's Prime Pages. Goldbach conjecture verification, Tomás Oliveira e Silva's distributed computer search. Additive number theory Analytic number theory Conjectures about prime numbers Unsolved problems in number theory Hilbert's problems