All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Previous Page 3 Next

Displaying 41 – 60 of 330

Showing per page

Contre-exemples au principe de Hasse pour les courbes de Fermat

Alain Kraus (2016)

Acta Arithmetica

Let p be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over ℚ, given by equations a x p + b y p + c z p = 0 , with respect to the local-global Hasse principle. It is conjectured that there exist infinitely many Fermat curves of exponent p which are counterexamples to the Hasse principle. This is a consequence of the abc-conjecture if p ≥ 5. Using a cyclotomic approach due to H. Cohen and Chebotarev’s density theorem, we obtain a partial result towards this conjecture, by...

Cubic forms, powers of primes and the Kraus method

Andrzej Dąbrowski, Tomasz Jędrzejak, Karolina Krawciów (2012)

Colloquium Mathematicae

We consider the Diophantine equation ( x + y ) ( x ² + B x y + y ² ) = D z p , where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D’s, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).

Diophantine equations after Fermat’s last theorem

Samir Siksek (2009)

Journal de Théorie des Nombres de Bordeaux

These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions:(i)Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem?(ii)Is it useful to combine this approach with traditional approaches to Diophantine equations: Diophantine approximation, arithmetic geometry, ...?

Diophantine equations and class number of imaginary quadratic fields

Zhenfu Cao, Xiaolei Dong (2000)

Discussiones Mathematicae - General Algebra and Applications

Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and μ i ∈ - 1 , 1 ( i = 1 , 2 ) , and let h ( - 2 1 - e D ) ( e = 0 o r 1 ) denote the class number of the imaginary quadratic field ℚ ( √ ( - 2 1 - e D ) ) . In this paper, we give the all-positive integer solutions of the Diophantine equation Ax² + μ₁B = K((Ay² + μ₂B)/K)ⁿ, 2 ∤ n, n > 1 and we prove that if D > 1, then h ( - 2 1 - e D ) ≡ 0 ( m o d n ) , where D, and n satisfy k ⁿ - 2 e + 1 = D x ² , x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.

Diophantine equations with Euler polynomials

Dijana Kreso, Csaba Rakaczki (2013)

Acta Arithmetica

We determine decomposition properties of Euler polynomials and using a strong result relating polynomial decomposition and diophantine equations in two separated variables, we characterize those g(x) ∈ ℚ [x] for which the diophantine equation - 1 k + 2 k - ⋯ + ( - 1 ) x x k = g ( y ) with k ≥ 7 may have infinitely many integer solutions. Apart from the exceptional cases we list explicitly, the equation has only finitely many integer solutions.

Currently displaying 41 – 60 of 330