�ber die Darstellung der Zahlen durch die bin�ren kubischen Formen von negativer Diskriminante

Delaunay, B.

doi:10.1007/bf01246394

Cited by 29 publications

(7 citation statements)

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“…Now, by a principle due to Lagrange [6], of the corresponding equation is bounded by an absolute constant, s was shewn by Delauney [3] for D > 0 and by Siegel [10] for D < 0 (their discriminant was of opposite sign to ours) ; while, if f i be reducible, then plainly there are at most four Solutions because D Ö 0. Therefore where the summation is over a representative System of forms U(x,y) and where r(k 2 ) = r O (k 2 ) is the number of Solutions in integers x, y (no longer necessarily relatively prime) of J(x,y) = k 2 .…”

Section: Binary Cubic Forms and The Estimation Of P 3 (Ar)mentioning

confidence: 99%

On the square-free values of cubic polynomials.

1968

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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IntroductionThe problem of the representation of power-free integers by integral polynomials appears to have been first considered by Nageil [8], who shewed in 1922 that an irreducible polynomial f(n) of degree r > 2 is Z-free (i. e. not divisible by an Z-th power other than 1) for infinitely many n provided / ^ r. This work was subsequently refined and developed by a number of writers, among whom we may mention Ricci, Estermann, Atkinson and Lord Gherwell, and Erdös. Using a sieve method Ricci [9], for example, proved in 1933 that, if N (x) -N (/, /) be the number of positive integers n not exceeding with the property that f(n) be /-free, then the asymptotic formulaäs -> oo, is valid for / > r. Estermann [5] had, however, already derived by another method a similar but more precise asymptotic formula for N(#) in the special case for which f(n) = n 2 + k and / = 2, while much later Atkinson and Lord Cherwell [1], independently of Estermann, proved an analogous formula for the less restricted special case f(n) = n r + k and l -r. Several other papers also appeared between the publication of Ricci's paper and 1953, but these, though of interest in many respects, did not substantially contribute to the solution of the main problem because they did not relate to the difficult case l < r. In the latter year Erdös [4] made a significant advance by proving in a characteristically ingenious manner that NagelPs result is true when l -r -l (r > 2) provided that the obvious necessary condition that f(n) have no (r -l)th power fixed divisors hold. His method, however, did not provide an asymptotic formula for N(#), nor did it shew that the integers n for which f(n) is (r -l)-free had positive upper or lower densities. Indeed, äs Erdös observed, the state of knowledge about the case l < r was still quite incomplete since it was not even known whether a particular quartic polynomial such äs r£ + 2 could be infinitely often square-free or whether for l = 2 Ricci's asymptotic formula held for any cubics at all 1 ).The purpose of this paper is to prove that the asymptotic formula z \ Iog 8 # !) A closely allied problem is that of the representation of large numbers äs the sum of an rth power and an Z-free number. A proof of an asymptotic formula for the number of representations has been attempted by Subhankulov and Moisezon without restriction on r and l (Izv.

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Section: Binary Cubic Forms and The Estimation Of P 3 (Ar)mentioning

confidence: 99%

On the square-free values of cubic polynomials.

1968

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…It remains to obtain an upper bound for the number of solutions of an equation of type (5) in which c has been specified. An important result of B. Delauney [2] and T. Nagell [3] states that the Diophantine equation 7pX3 + qX2Y + rXY2 + sY3 = 1 has at most five solutions, provided that the discriminant…”

Section: B2mentioning

confidence: 99%

Some remarks on the Diophantine equation 𝑥³+𝑦³+𝑧³=𝑥+𝑦+𝑧

Edgar¹

1965

Proc. Amer. Math. Soc.

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“…If f (x, y) is an irreducible binary cubic form with negative discriminant, Delauney [10] and Nagell [30] showed that the equation f (x, y) = 1 has at most five integer solutions (x, y). Now if its discriminant is positive, then Evertse [13] showed that the equation f (x, y) = 1 has at most twelve integer solutions (x, y).…”

Section: Introductionmentioning

confidence: 99%

On cubic Thue equations and the indices of algebraic integers in cubic fields

Bayad

Seddik

2019

APPL ANAL DISCRETE M

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Let F (x, y) = ax 3 + bx 2 y + cxy 2 + dy 3 ∈ Z[x, y] be an irreducible cubic form. In this paper, we investigate arithmetic properties of the common indices of algebraic integers in cubic fields. For each integer k such that v2(k) ≡ 0 (mod 3) and 2v2(−2b 3 − 27a 2 d + 9abc) = 3v2(b 2 − 3ac), we prove that the cubic Thue equation F (x, y) = k has no solution (x, y) ∈ Z 2. As application, we construct parametrized families of twisted elliptic curves E : ax 3 + bx 2 + cx + d = ey 2 without integer points (x, y).

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�ber die Darstellung der Zahlen durch die bin�ren kubischen Formen von negativer Diskriminante

Cited by 29 publications

References 0 publications

On the square-free values of cubic polynomials.

On the square-free values of cubic polynomials.

Some remarks on the Diophantine equation 𝑥³+𝑦³+𝑧³=𝑥+𝑦+𝑧

On cubic Thue equations and the indices of algebraic integers in cubic fields

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