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pro vyhledávání: '"Everest, Graham"'
We show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K, a positive
Externí odkaz:
http://arxiv.org/abs/1012.4878
Autor:
Everest, Graham, Ward, Thomas
Publikováno v:
Amer. Math. Monthly 118(7), 584-598 (2011)
Problems related to the existence of integral and rational points on cubic curves date back at least to Diophantus. A significant step in the modern theory of these equations was made by Siegel, who proved that a non-singular plane cubic equation has
Externí odkaz:
http://arxiv.org/abs/1005.0315
On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the rational point i
Externí odkaz:
http://arxiv.org/abs/1003.2131
Autor:
Everest, Graham, Griffiths, Jonny
This article has been written for an educational magazine whose target audience consists of students and teachers of mathematics in universities, colleges and schools. It concerns a notion of duality between rectangles. A proof is given that only fin
Externí odkaz:
http://arxiv.org/abs/0906.3096
Autor:
Everest, Graham, Mahe, Valery
Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is placed upo
Externí odkaz:
http://arxiv.org/abs/0803.0700
An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over the rationa
Externí odkaz:
http://arxiv.org/abs/0712.2696
Autor:
Everest, Graham, Eisentraeger, Kirsten
Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert's Tenth Problem is undecidable. This method further develops that of Poon
Externí odkaz:
http://arxiv.org/abs/0707.1485
We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u^3+v^3=m, with m cube-free, all the terms beyond the first have a primitive divisor.
Comment: 33 pages, 4 figures
Comment: 33 pages, 4 figures
Externí odkaz:
http://arxiv.org/abs/math/0703553
Autor:
Everest, Graham, Harman, Glyn
We study primitive divisors of terms of the sequence P_n=n^2+b, for a fixed integer b which is not a negative square. It seems likely that the number of terms with a primitive divisor has a natural density. This seems to be a difficult problem. We su
Externí odkaz:
http://arxiv.org/abs/math/0701234
Publikováno v:
Math. Intelligencer, 31, No. 3, 13-17 (2009)
We show how the Binomial Theorem can be used to continue the Riemann Zeta Function to the left hand half-plane. This method yields the explicit values of the function at non-positive integers in terms of the Bernoulli numbers.
Comment: Extra ref
Comment: Extra ref
Externí odkaz:
http://arxiv.org/abs/math/0610108