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In number theory, the Erdős–Straus conjecture states that for all integers ''n'' ≥ 2, the rational number 4/''n'' can be expressed as the sum of three unit fractions. Paul Erdős and Ernst G. Straus formulated the conjecture in 1948.〔See, e.g., . Note however that the earliest published reference to it appears to be .〕 It is one of many conjectures by Erdős. More formally, the conjecture states that, for every integer ''n'' ≥ 2, there exist positive integers ''x'', ''y'', and ''z'' such that : Some researchers additionally require these integers to be distinct from each other, while others allow them to be equal; if they are distinct then these unit fractions form an Egyptian fraction representation of the number 4/''n''. For instance, for ''n'' = 5, there are two solutions: : The restriction that ''x'', ''y'', and ''z'' be positive is essential to the difficulty of the problem, for if negative values were allowed the problem could be solved trivially. Also, if ''n'' is a composite number, ''n'' = ''pq'', then an expansion for 4/''n'' could be found immediately from an expansion for 4/''p'' or 4/''q''. Therefore, if a counterexample to the Erdős–Straus conjecture exists, the smallest ''n'' forming a counterexample would have to be a prime number, and it can be further restricted to one of six infinite arithmetic progressions modulo 840.〔.〕 Computer searches have verified the truth of the conjecture up to ''n'' ≤ 1014,〔 but proving it for all ''n'' remains an open problem. As long as ''n'' ≥ 3, it does not matter whether the three natural numbers x, y, z are required to be distinct or not: if there exists a solution with any three integers ''x'', ''y'', and ''z'' then there exists a solution with distinct integers. In the case ''n'' = 2, however, the only solution is 4/2 = 1/2 + 1/2 + 1/1, up to permutation of the summands. ==Background== The search for expansions of rational numbers as sums of unit fractions dates to the mathematics of ancient Egypt, in which Egyptian fraction expansions of this type were used as a notation for recording fractional quantities. The Egyptians produced tables such as the Rhind Mathematical Papyrus 2/n table of expansions of fractions of the form 2/''n'', most of which use either two or three terms. Egyptian fractions typically have an additional constraint, that all of the unit fractions be distinct from each other, but for the purposes of the Erdős–Straus conjecture this makes no difference: if 4/''n'' can be expressed as a sum of at most three unit fractions, it can also be expressed as a sum of at most three distinct unit fractions. The greedy algorithm for Egyptian fractions, first described in 1202 by Fibonacci in his book Liber Abaci, finds an expansion in which each successive term is the largest unit fraction that is no larger than the remaining number to be represented. For fractions of the form 2/''n'' or 3/''n'', the greedy algorithm uses at most two or three terms respectively. More generally, it can be shown that a number of the form 3/''n'' has a two-term expansion if and only if ''n'' has a factor congruent to 2 modulo 3, and requires three terms in any expansion otherwise.〔.〕 Thus, for the numerators 2 and 3, the question of how many terms are needed in an Egyptian fraction is completely settled, and fractions of the form 4/''n'' are the first case in which the worst-case length of an expansion remains unknown. The greedy algorithm produces expansions of length two, three, or four depending on the value of ''n'' modulo 4; when ''n'' is congruent to 1 modulo 4, the greedy algorithm produces four-term expansions. Therefore, the worst-case length of an Egyptian fraction of 4/''n'' must be either three or four. The Erdős–Straus conjecture states that, in this case, as in the case for the numerator 3, the maximum number of terms in an expansion is three. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Erdős–Straus conjecture」の詳細全文を読む スポンサード リンク
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