Erdős–Straus conjecture (original) (raw)
Die zahlentheoretische Erdős-Straus-Vermutung (nach den Mathematikern Paul Erdős und Ernst Gabor Straus) besagt, dass stets einer Summe von drei positiven Stammbrüchen entspricht. Sie wurde im Jahr 1948 aufgestellt und ist eine von vielen Vermutungen von Paul Erdős.
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| dbo:abstract | Die zahlentheoretische Erdős-Straus-Vermutung (nach den Mathematikern Paul Erdős und Ernst Gabor Straus) besagt, dass stets einer Summe von drei positiven Stammbrüchen entspricht. Sie wurde im Jahr 1948 aufgestellt und ist eine von vielen Vermutungen von Paul Erdős. (de) Unsolved problem in mathematics: Does have a positive integer solution for every integer ? (more unsolved problems in mathematics) The Erdős–Straus conjecture is an unproven statement in number theory. The conjecture is that, for every integer that is 2 or more, there exist positive integers , , and for which In other words, the number can be written as a sum of three positive unit fractions. The conjecture is named after Paul Erdős and Ernst G. Straus, who formulated it in 1948, but it is connected to much more ancient mathematics; sums of unit fractions, like the one in this problem, are known as Egyptian fractions, because of their use in ancient Egyptian mathematics. The Erdős–Straus conjecture is one of many conjectures by Erdős, and one of many unsolved problems in mathematics concerning Diophantine equations. Although a solution is not known for all values of n, infinitely many values in certain infinite arithmetic progressions have simple formulas for their solution, and skipping these known values can speed up searches for counterexamples. Additionally, these searches need only consider values of that are prime numbers, because any composite counterexample would have a smaller counterexample among its prime factors. Computer searches have verified the truth of the conjecture up to . If the conjecture is reframed to allow negative unit fractions, then it is known to be true. Generalizations of the conjecture to fractions with numerator 5 or larger have also been studied. (en) La conjecture d'Erdős-Straus énonce que tout nombre rationnel de la forme , avec n entier supérieur ou égal à 2, peut être écrit comme somme de trois fractions unitaires, c'est-à-dire qu'il existe trois entiers naturels non nuls et tels que : Louis Mordell a montré que pour la conjecture est vraie. (fr) La congettura di Erdős-Straus afferma che per ogni intero , il numero razionale 4/n si può scrivere come somma di tre frazioni unitarie, ossia esistono tre interi positivi , e tali che La somma di queste frazioni unitarie è una rappresentazione come frazione egiziana del numero 4/n. Ad esempio, per n = 1801, esiste una soluzione con x = 451, y = 295364 e z = 3249004: Moltiplicando entrambi i membri dell'equazione per nxyz si trova la l'equazione diofantea equivalente 4xyz=n(xy+xz+yz). La restrizione di x, y, e z ai numeri positivi è cruciale per la difficoltà del problema, dal momento che, se i valori negativi fossero ammessi, il problema potrebbe essere risolto banalmente da una delle due identità 4/(4k+1) = 1/k - 1/k(4k+1) e 4/(4k-1) = 1/k + 1/k(4k-1). Se n è un numero composto, n = pq, allora si potrebbe trovare immediatamente una soluzione come somma di frazioni egiziane per 4/n dalla soluzione per 4/p o per 4/q. Pertanto, se esistono controesempi alla congettura di Erdős–Straus, il più piccolo deve necessariamente essere un numero primo. Paul Erdős e formularono la congettura nel 1948 (vedi, ad esempio, Elsholtz) ma il primo riferimento divulgato sembra essere una pubblicazione di Erdős del 1950. (it) 수론에서 1948년, 에르되시 팔과 (Ernst G. Straus)가 추측에 사용한 공식이다. 에르되시-스트라우스 추측(Erdős–Straus conjecture)이라고 한다. 정수에 대해서 n ≥ 2일때, 자연수 x, y, z의 해가 언제나 존재한다라고하는 것에 대한 추측이다. 예로, n = 5는 다음과 같은 2개의 해가 존재한다. 2013년, 테렌스 타오가 크리스티안 엘숄츠(Christian Elsholtz)와 함께 이 문제에대한 추측상의 출현 수 세기에 대한 논문을 발표했다. 수학의 미해결 문제이다. 또한 이것은 이라는 피타고라스의 정리의 연장선상에 있는 이라는 디오판토스의 방정식의 분수형태의 변형과 관계있다. (ko) Het vermoeden van Erdős-Straus is een nog niet bewezen vermoeden uit de getaltheorie dat stelt dat door welk getal groter dan 1 je 4 ook deelt, het quotiënt altijd de som van drie stambreuken is. Paul Erdős en stelden het vermoeden op in 1948. Het is een van de vele vermoedens van Erdős. Formeel luidt het vermoeden:voor iedere gehele geldt dat er positieve getallen zijn, zo dat (nl) Гипотеза Эрдёша — Штрауса — теоретико-числовая гипотеза, согласно которой для всех целых чисел рациональное число может быть представлено в виде суммы трёх аликвотных дробей (дробей с единицей в числителе), то есть существует три положительных целых числа , и , таких что: . Сформулирована в 1948 году Палом Эрдёшом и Эрнстом Штраусом. Перебором на компьютере проверено выполнение гипотезы для всех чисел вплоть до , но доказательство для всех остаётся по состоянию на 2015 год открытой проблемой. (ru) Inom talteori är Erdős–Straus förmodan en förmodan som säger att för alla heltal n ≥ 2 kan talet 4/n skrivas som summan av reciprokerna av tre positiva heltal. Paul Erdős och formulerade förmodandet år 1948. (sv) 歐德斯-史特勞斯猜想（Erdős–Straus conjecture），簡稱歐德斯猜想，是由匈牙利犹太数学家保罗·埃尔德什與德裔美國數學家於1948年共同提出的數論猜想，其陳述为： 对于任何一个大于1的整数，都有 。其中, , 为正整数。 例如，若n = 1801，則存在一組 x = 451、y = 295364、z = 3249004 的解，使得 在基本式子中，只需考慮 n = p 為素數的情況，因為若 成立，則對於大於 1 的整數 m 也會成立。 計算機已經驗證到 n ≤ 1014 的情況，但此猜想還是有待證明。 (zh) |
| dbo:wikiPageExternalLink | https://terrytao.files.wordpress.com/2011/07/egyptian-count13.pdf http://www.ics.uci.edu/~eppstein/numth/egypt/smallnum.html https://cms.math.ca/publications/crux/issue/%3Fvolume=30&issue=1 http://www.renyi.hu/~p_erdos/1950-02.pdf https://www.math.ucsd.edu/~ronspubs/13_03_Egyptian.pdf |
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| rdfs:comment | Die zahlentheoretische Erdős-Straus-Vermutung (nach den Mathematikern Paul Erdős und Ernst Gabor Straus) besagt, dass stets einer Summe von drei positiven Stammbrüchen entspricht. Sie wurde im Jahr 1948 aufgestellt und ist eine von vielen Vermutungen von Paul Erdős. (de) La conjecture d'Erdős-Straus énonce que tout nombre rationnel de la forme , avec n entier supérieur ou égal à 2, peut être écrit comme somme de trois fractions unitaires, c'est-à-dire qu'il existe trois entiers naturels non nuls et tels que : Louis Mordell a montré que pour la conjecture est vraie. (fr) 수론에서 1948년, 에르되시 팔과 (Ernst G. Straus)가 추측에 사용한 공식이다. 에르되시-스트라우스 추측(Erdős–Straus conjecture)이라고 한다. 정수에 대해서 n ≥ 2일때, 자연수 x, y, z의 해가 언제나 존재한다라고하는 것에 대한 추측이다. 예로, n = 5는 다음과 같은 2개의 해가 존재한다. 2013년, 테렌스 타오가 크리스티안 엘숄츠(Christian Elsholtz)와 함께 이 문제에대한 추측상의 출현 수 세기에 대한 논문을 발표했다. 수학의 미해결 문제이다. 또한 이것은 이라는 피타고라스의 정리의 연장선상에 있는 이라는 디오판토스의 방정식의 분수형태의 변형과 관계있다. (ko) Het vermoeden van Erdős-Straus is een nog niet bewezen vermoeden uit de getaltheorie dat stelt dat door welk getal groter dan 1 je 4 ook deelt, het quotiënt altijd de som van drie stambreuken is. Paul Erdős en stelden het vermoeden op in 1948. Het is een van de vele vermoedens van Erdős. Formeel luidt het vermoeden:voor iedere gehele geldt dat er positieve getallen zijn, zo dat (nl) Гипотеза Эрдёша — Штрауса — теоретико-числовая гипотеза, согласно которой для всех целых чисел рациональное число может быть представлено в виде суммы трёх аликвотных дробей (дробей с единицей в числителе), то есть существует три положительных целых числа , и , таких что: . Сформулирована в 1948 году Палом Эрдёшом и Эрнстом Штраусом. Перебором на компьютере проверено выполнение гипотезы для всех чисел вплоть до , но доказательство для всех остаётся по состоянию на 2015 год открытой проблемой. (ru) Inom talteori är Erdős–Straus förmodan en förmodan som säger att för alla heltal n ≥ 2 kan talet 4/n skrivas som summan av reciprokerna av tre positiva heltal. Paul Erdős och formulerade förmodandet år 1948. (sv) 歐德斯-史特勞斯猜想（Erdős–Straus conjecture），簡稱歐德斯猜想，是由匈牙利犹太数学家保罗·埃尔德什與德裔美國數學家於1948年共同提出的數論猜想，其陳述为： 对于任何一个大于1的整数，都有 。其中, , 为正整数。 例如，若n = 1801，則存在一組 x = 451、y = 295364、z = 3249004 的解，使得 在基本式子中，只需考慮 n = p 為素數的情況，因為若 成立，則對於大於 1 的整數 m 也會成立。 計算機已經驗證到 n ≤ 1014 的情況，但此猜想還是有待證明。 (zh) Unsolved problem in mathematics: Does have a positive integer solution for every integer ? (more unsolved problems in mathematics) The Erdős–Straus conjecture is an unproven statement in number theory. The conjecture is that, for every integer that is 2 or more, there exist positive integers , , and for which In other words, the number can be written as a sum of three positive unit fractions. If the conjecture is reframed to allow negative unit fractions, then it is known to be true. Generalizations of the conjecture to fractions with numerator 5 or larger have also been studied. (en) La congettura di Erdős-Straus afferma che per ogni intero , il numero razionale 4/n si può scrivere come somma di tre frazioni unitarie, ossia esistono tre interi positivi , e tali che La somma di queste frazioni unitarie è una rappresentazione come frazione egiziana del numero 4/n. Ad esempio, per n = 1801, esiste una soluzione con x = 451, y = 295364 e z = 3249004: Paul Erdős e formularono la congettura nel 1948 (vedi, ad esempio, Elsholtz) ma il primo riferimento divulgato sembra essere una pubblicazione di Erdős del 1950. (it) |
| rdfs:label | Erdős-Straus-Vermutung (de) Erdős–Straus conjecture (en) Conjecture d'Erdős-Straus (fr) Congettura di Erdős-Straus (it) 에르되시-스트라우스 추측 (ko) Vermoeden van Erdős-Straus (nl) Гипотеза Эрдёша — Штрауса (ru) Erdős–Straus förmodan (sv) 歐德斯-史特勞斯猜想 (zh) |
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