On Isotropic Congruences in Elliptic Space.
Radu Rosca (1968)
Mathematische Zeitschrift
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Radu Rosca (1968)
Mathematische Zeitschrift
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Tsuneo Ishikawa (2006)
Acta Arithmetica
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Harlan Stevens (1962)
Mathematische Zeitschrift
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J.C.E. Dekker (1958)
Mathematische Zeitschrift
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L.J. Mordell (1933)
Mathematische Zeitschrift
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Tao Yan Zhao, Lily J. Jin, C. Gu (2016)
Open Mathematics
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Let p3(n) denote the number of 3-component multipartitions of n. Recently, using a 3-dissection formula for the generating function of p3(n), Baruah and Ojah proved that for n ≥ 0, p3(9n + 5) ≡ 0 (mod 33) and p3 (9n + 8) ≡ 0 (mod 34). In this paper, we prove several congruences modulo powers of 3 for p3(n) by using some theta function identities. For example, we prove that for n ≥ 0, p3 (243n + 233) ≡ p3 (729n + 638) ≡ 0 (mod 310).
John B. Cosgrave, Karl Dilcher (2010)
Acta Arithmetica
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J. Lawson, Bernard Madison (1971)
Mathematische Zeitschrift
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J.M. Gandhi (1965)
Monatshefte für Mathematik
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Olcay Karaatlı (2016)
Acta Arithmetica
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Let Vₙ(P,Q) denote the generalized Lucas sequence with parameters P and Q. For all odd relatively prime values of P and Q such that P² + 4Q > 0, we determine all indices n such that Vₙ(P,Q) = 7kx² when k|P. As an application, we determine all indices n such that the equation Vₙ = 21x² has solutions.
M. M., Slotnick (1928)
Mathematische Zeitschrift
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Karl Rubin (1983)
Inventiones mathematicae
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L. Carlitz, W.A. Al-Salam (1958)
Monatshefte für Mathematik
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Stanley Burris (1971)
Colloquium Mathematicae
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Glenn J. Fox (2002)
Journal de théorie des nombres de Bordeaux
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We illustrate how a particular expression, involving the generalized Bernoulli polynomials, satisfies systems of congruence relations if and only if a similar expression, involving the generalized Bernoulli numbers, satisfies the same congruence relations. These congruence relations include the Kummer congruences, and recent extensions of the Kummer congruences provided by Gunaratne.