In modular arithmetic, Thue's lemma roughly states that every modular integer may be represented by a "modular fraction" such that the numerator and the denominator have absolute values not greater than the square root of the modulus. More precisely, for every pair of integers (a, m) with m > 1, given two positive integers X and Y such that X ≤ m < XY, there are two integers x and y such that and Usually, one takes X and Y equal to the smallest integer greater than the square root of m, but the general form is sometimes useful, and makes the uniqueness theorem (below) easier to state.
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| - Lema de Thue (ca)
- Lemma von Thue (de)
- Lemme de Thue (fr)
- Lemma di Thue (it)
- 투에 보조정리 (ko)
- Thue's lemma (en)
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| - El lema de Thue és un resultat en teoria de nombres que afirma que si p és un nombre primer de la forma 4k + 1, llavors existeixen dos únics nombres enters a i b, amb 0 < a < b, tals que p = a² + b². A més, si un nombre n es pot escriure com a suma de dos quadrats de dues formes diferents, llavors n és un nombre compost. (ca)
- Das Lemma von Thue, bei manchen Autoren auch Satz von Thue genannt, ist ein Lehrsatz der Elementaren Zahlentheorie, eines Teilgebiets der Mathematik. Es geht auf den norwegischen Mathematiker Axel Thue zurück und spielt eine Rolle bei Untersuchungen zu diophantischen Gleichungen. Der Beweis beruht auf dem dirichletschen Schubfachprinzip. (de)
- En arithmétique modulaire, le lemme de Thue établit que tout entier modulo m peut être représenté par une « fraction modulaire » dont le numérateur et le dénominateur sont, en valeur absolue, majorés par la racine carrée de m. La première démonstration, attribuée à Axel Thue, utilise le principe des tiroirs. Appliqué à un entier m modulo lequel –1 est un carré (en particulier à un nombre premier m congru à 1 modulo 4) et à un entier a tel que a2 + 1 ≡ 0 mod m, ce lemme fournit une expression de m comme somme de deux carrés premiers entre eux. (fr)
- Il lemma di Thue, chiamato così dal matematico norvegese Axel Thue, è un lemma della teoria dei numeri che afferma che, per ogni numero primo p e per ogni intero , la congruenza (dove indica l'operazione modulo). ammette una soluzione tale che . Può essere usato per dimostrare il teorema di Fermat sulle somme di due quadrati. (it)
- 수론에서 투에 보조정리(-補助定理, 영어: Thue's lemma)는 일차 합동 방정식이 다소 작은 해를 가질 충분 조건을 제시하는 정리이다. 비둘기집 원리의 수론에서의 한 가지 응용이다. 페르마 두 제곱수 정리의 증명에 사용된다. (ko)
- In modular arithmetic, Thue's lemma roughly states that every modular integer may be represented by a "modular fraction" such that the numerator and the denominator have absolute values not greater than the square root of the modulus. More precisely, for every pair of integers (a, m) with m > 1, given two positive integers X and Y such that X ≤ m < XY, there are two integers x and y such that and Usually, one takes X and Y equal to the smallest integer greater than the square root of m, but the general form is sometimes useful, and makes the uniqueness theorem (below) easier to state. (en)
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| - El lema de Thue és un resultat en teoria de nombres que afirma que si p és un nombre primer de la forma 4k + 1, llavors existeixen dos únics nombres enters a i b, amb 0 < a < b, tals que p = a² + b². A més, si un nombre n es pot escriure com a suma de dos quadrats de dues formes diferents, llavors n és un nombre compost. (ca)
- Das Lemma von Thue, bei manchen Autoren auch Satz von Thue genannt, ist ein Lehrsatz der Elementaren Zahlentheorie, eines Teilgebiets der Mathematik. Es geht auf den norwegischen Mathematiker Axel Thue zurück und spielt eine Rolle bei Untersuchungen zu diophantischen Gleichungen. Der Beweis beruht auf dem dirichletschen Schubfachprinzip. (de)
- En arithmétique modulaire, le lemme de Thue établit que tout entier modulo m peut être représenté par une « fraction modulaire » dont le numérateur et le dénominateur sont, en valeur absolue, majorés par la racine carrée de m. La première démonstration, attribuée à Axel Thue, utilise le principe des tiroirs. Appliqué à un entier m modulo lequel –1 est un carré (en particulier à un nombre premier m congru à 1 modulo 4) et à un entier a tel que a2 + 1 ≡ 0 mod m, ce lemme fournit une expression de m comme somme de deux carrés premiers entre eux. (fr)
- In modular arithmetic, Thue's lemma roughly states that every modular integer may be represented by a "modular fraction" such that the numerator and the denominator have absolute values not greater than the square root of the modulus. More precisely, for every pair of integers (a, m) with m > 1, given two positive integers X and Y such that X ≤ m < XY, there are two integers x and y such that and Usually, one takes X and Y equal to the smallest integer greater than the square root of m, but the general form is sometimes useful, and makes the uniqueness theorem (below) easier to state. The first known proof is attributed to Axel Thue who used a pigeonhole argument. It can be used to prove Fermat's theorem on sums of two squares by taking m to be a prime p that is congruent to 1 modulo 4 and taking a to satisfy a2 + 1 = 0 mod p. (Such an "a" is guaranteed for "p" by Wilson's theorem.) (en)
- Il lemma di Thue, chiamato così dal matematico norvegese Axel Thue, è un lemma della teoria dei numeri che afferma che, per ogni numero primo p e per ogni intero , la congruenza (dove indica l'operazione modulo). ammette una soluzione tale che . Può essere usato per dimostrare il teorema di Fermat sulle somme di due quadrati. (it)
- 수론에서 투에 보조정리(-補助定理, 영어: Thue's lemma)는 일차 합동 방정식이 다소 작은 해를 가질 충분 조건을 제시하는 정리이다. 비둘기집 원리의 수론에서의 한 가지 응용이다. 페르마 두 제곱수 정리의 증명에 사용된다. (ko)
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