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By Ernie Croot, Andrew Granville, Robin Pemantle, Prasad Tetali (auth.), Alfred J. van der Poorten, Andreas Stein (eds.)

This booklet constitutes the refereed court cases of the eighth foreign Algorithmic quantity idea Symposium, ANTS 2008, held in Banff, Canada, in may well 2008.

The 28 revised complete papers provided including 2 invited papers have been rigorously reviewed and chosen for inclusion within the ebook. The papers are geared up in topical sections on elliptic curves cryptology and generalizations, mathematics of elliptic curves, integer factorization, K3 surfaces, quantity fields, element counting, mathematics of functionality fields, modular types, cryptography, and quantity theory.

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Extra info for Algorithmic Number Theory: 8th International Symposium, ANTS-VIII Banff, Canada, May 17-22, 2008 Proceedings

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48) We now establish that γM,k (u) < ∞ except perhaps when M = k = ∞: We have 0 ≤ AM (z) ≤ log M for all z, so that u < γM,k (u) ≤ M u for all u; in particular γM,k (u) < ∞ if M < ∞. ; in particular γ∞,k (u) < ∞. As M, k → ∞, the fixed point γM,k (u) increases to the fixed point γ(u) of the map z → ueA(z) , or to ∞ if there is no such fixed point, in which case we write γ(u) = ∞. By comparing this with (44) we see that γ(u) = uf (u). In [4] we show that this map has a fixed point if and only if η u ≤ e−γ .

1) for any odd n. 1). This approach is another version of an idea of Legendre as modified by Dujardin (see Dickson [6, p. 416]). 6) will yield all of the solutions of X 2 − DY 2 = N. As we have already mentioned there are only a finite number of such pairs. There may be none at all. 3) becomes X 2−D Y2 =N . We may therefore assume with no loss of generality that gcd(D, N ) is squarefree. A New Look at an Old Equation 41 In order to proceed further we will make use of some results from the theory of real quadratic number fields and some associated algorithms.

As M, k → ∞, the fixed point γM,k (u) increases to the fixed point γ(u) of the map z → ueA(z) , or to ∞ if there is no such fixed point, in which case we write γ(u) = ∞. By comparing this with (44) we see that γ(u) = uf (u). In [4] we show that this map has a fixed point if and only if η u ≤ e−γ . Otherwise γ(u) = ∞ for u > e−γ so that 0 γ(u) u du = ∞ > 1 for any −γ η>e . One might ask how the variables m, M, k, u relate to our problem? We are looking at the possible pseudosmooths (that is integers which are a y0 -smooth times a square) composed of products of aj with j ≤ uJ0 .

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