Odd Greedy Expansion
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In number theory, the odd greedy expansion problem concerns a method for forming Egyptian fractions in which all denominators are odd. However, there is a simpler greedy algorithm that has successfully found Egyptian fractions in which all denominators are odd for all instances x/y (with odd y) on which it has been tested: let u be the least odd number that is greater than or equal to y/x, include the fraction 1/u in the expansion, and continue in the same way with th...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In number theory, the odd greedy expansion problem concerns a method for forming Egyptian fractions in which all denominators are odd. However, there is a simpler greedy algorithm that has successfully found Egyptian fractions in which all denominators are odd for all instances x/y (with odd y) on which it has been tested: let u be the least odd number that is greater than or equal to y/x, include the fraction 1/u in the expansion, and continue in the same way with the remaining fraction x/y - 1/u. This method is called the odd greedy algorithm and the expansions it creates are called odd greedy expansions. Stein, Selfridge, Graham, and others have posed the question of whether the odd greedy algorithm terminates with a finite expansion for every x/y with y odd (Guy 1981). As of 2006, this question remains open.
