, On the simplicity of an axiom system for plane Euclidean geometry, Demonstratio Mathematica, vol. 30 (1997), pp. 509--512. Home/Search Document Not in Database Summary Related Articles Check |
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Searching for Primitive Roots in Finite Fields - Shoup (1992) (19 citations) (Correct)
....Jeff Lagarias, and Andrew Odlyzko for helpful discussions, and Hendrik Lenstra for comments on earlier versions of this paper that led to the simplification of several proofs. 2. Preliminaries As does Wang, we shall make use of a combinatorial sieve. However, we will use a sieve due to Iwaniec [17] that is easier to apply and gives sharper upper bounds. Iwaniec specifically considers a problem known as Jacobsthal s problem, which is to estimate for a given r the maximum length C(r) of a sequence of consecutive integers each divisible by one of r arbitrarily chosen primes. Iwaniec proves ....
....Assume that q 1 Delta Delta Delta q r . Let p 1 Delta Delta Delta p r be the first r primes, and let P = p 1 Delta Delta Delta p r . Let f n : n j Pg be a set of real numbers, and let oe n = P mjn m for n j P . Assume that oe n P mjn (m) for all n j P . Lemma 1 in [17] can be easily generalized to obtain T A r Y i=1 (1 Gamma 1 q i )G 1 Gamma BG 2 ; 2) where G 1 = X njP oe n Q p i jn (p i Gamma 1) and G 2 = X njP j n j: Let z = p r , and let y satisfy z 2 y z 4 . By choosing the numbers n appropriately (see the definition on p. 5 ....
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H. Iwaniec. On the problem of Jacobsthal. Demonstratio Mathematica, 11:225--231, 1978.
The Modulo N Extended GCD Problem for Polynomials - Mulders, Storjohann (Correct)
....(N) be the minimal t 2 N such that for all a; b 2 F [x] with gcd(a; b; N) 1, there exists an f 2 F [x] of degree t such that gcd(a fb; N) 1. 1 Note that [9] contains an error. Fact 1 in Section 2 is an incorrect quote due to the author of [9] A correct bound is g(N) O(r 2 ) see [4]) Slight modifications of the algorithms in [9] admit the same asymptotic bounds for the running time and size of the output. A correction is in progress. Lemma 1 Let a; b; N 2 F [x] such that N 6= 0 and gcd(a; b; N) 1, and let fp1 ; pkg be the set of all irreducible monic divisors ....
....gF (N) hF (N ) ffl We see that the function gF is similar to the wellknown Jacobsthal function g from number theory (see [5] 8] which can be defined by g(n) minft 2 N j 8a2N 9 i2f1; tg : gcd(a i; n) 1g, for n 2 N . We therefore call gF the polynomial Jacobsthal function for F . In [4] it is proven that g(n) O(k 2 ) where k is the number of prime divisors of n. From now on let N 2 F [x] nf0g. Let I(N) be the set of irreducible monic divisors of N . We have already seen (Lemma 3) that when F contains more than #I(N) elements, then gF (N) 0. So when F is an infinite ....
Iwaniec, H. On the problem of Jacobsthal. Demonstratio Mathematica 11, 1 (1978), 225---231.
Tarski's System of Geometry - Tarski, Givant (1999) (1 citation) Self-citation (Geometry) (Correct)
....[20] gives a set of axioms equivalent to CG (n) that uses exactly n 2 distinct variables. The same paper gives sets of axioms equivalent to PG (2) and to CG (2) that use only six distinct variables and five distinct variables respectively; it is shown in op. cit. and in Pambuccian [21] that these results are optimal: there are no equivalent sets of axioms that use fewer distinct variables. 8 Scott [30] writes: Though all details have not been completely checked by the author, it would seem that an adequate axiomatization of the theory E [elementary dimension free Euclidean ....
, On the simplicity of an axiom system for plane Euclidean geometry, Demonstratio Mathematica, vol. 30 (1997), pp. 509--512.
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