Ratios of congruent numbers
نویسندگان
چکیده
منابع مشابه
Some Families of Non-congruent Numbers
In this article we study the Tate-Shafarevich groups corresponding to 2-isogenies of the curve Ek : y 2 = x(x2 − k2) and construct infinitely many examples where these groups have odd 2-rank. Our main result is that among the curves Ek, where k = pl ≡ 1 mod 8 for primes p and l, the curves with rank 0 have density ≥ 1 2 .
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Mohammed Ben Alhocain, in an Arab manuscript of the 10th century, stated that the principal object of the theory of rational right triangles is to find a square that when increased or diminished by a certain number, m becomes a square [Dickson LE (1971) History of the Theory of Numbers (Chelsea, New York), Vol 2, Chap 16]. In modern language, this object is to find a rational point of infinite ...
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This improves an earlier result of the first author for k = n = 2, namely Theorem II below, which dealt with B2 -sequences. This notion arose from a paper of Sidon [6]. A strictly increasing sequence A of natural numbers a,, a2 . . . . is called a B2-sequence, if for any two pairs (i, j) + (k, m) of natural numbers i <j, k < m there holds aj a i + ak a„„ in short, if A has no double-differences...
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* Contributions from the Scripps Institution of Oceanography, New Series, No. 324. 'Ekman, V. W., Annalen Hydrographie u. Mar. Met., 34, 423-430 (1906). 2Stockmann, W., Comptes rendus (Doklady) l'Acad. sci. I'U.R.S.S., 52, 309-312 (1946). Fjeldstad, J. E., Archiv Math. Naturvid., 48, no. 6 (1946). 4 Montgomery, R. B., and Palmen, E., Jour. Marine Research, 3, 112-133 (1940). 6 Ekman, V. W., Ger...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2007
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa128-2-1