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In number theory, Znám's problem asks which sets of ''k'' integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other mathematicians had considered similar problems around the same time. One closely related problem drops the assumption of properness of the divisor, and will be called the improper Znám problem hereafter. One solution to the improper Znám problem is easily provided for any ''k'': the first ''k'' terms of Sylvester's sequence have the required property. showed that there is at least one solution to the (proper) Znám problem for each ''k'' ≥ 5. Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values. The Znám problem is closely related to Egyptian fractions. It is known that there are only finitely many solutions for any fixed ''k''. It is unknown whether there are any solutions to Znám's problem using only odd numbers, and there remain several other open questions. ==The problem== Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. That is, given ''k'', what sets of integers : are there, such that, for each ''i'', ''n''''i'' divides but is not equal to : A closely related problem concerns sets of integers in which each integer in the set is a divisor, but not necessarily a proper divisor, of one plus the product of the other integers in the set. This problem does not seem to have been named in the literature, and will be referred to as the improper Znám problem. Any solution to Znám's problem is also a solution to the improper Znám problem, but not necessarily vice versa. 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「Znám's problem」の詳細全文を読む スポンサード リンク
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