Abstract
In this paper, we study algebraic properties of lattice points of the arc on the conics x2 − dy2 = N especially for d = 1, which is the Fermat factorization equation that is the main idea of many important factorization methods like the quadratic field sieve, using arithmetical results of a particular hyperbola parametrization. As a result, we present a generalization of the forms, the cardinal, and the distribution of its lattice points over the integers. In particular, we prove that if (N − 6) ≡ 0 mod 4, Fermat’s method fails. Otherwise, in terms of cardinality, it has, respectively, 4, 8, 2(α + 1), , and lattice pointts if N is an odd prime, N = Na × Nb with Na and Nb being odd primes, with Na being prime, with pi being distinct primes, and with Ni being odd primes. These results are important since they provide further arithmetical understanding and information on the integer solutions revealing factors of N. These results could be particularly investigated for the purpose of improving the underlying integer factorization methods.
Highlights
We use the hyperbola parametrization introduced in [16] to study algebraic properties of lattice points and their distribution for Fermat’s factorization equation for which we find exact upper and lower bounds and we present the forms and cardinalities with a generalization of results for most of special cases of N, using results from the particular hyperbola parametrization
E article is organized as follows: (i) In Section 1, we give an introduction (ii) In Section 2, we present the particular hyperbola parametrization and related arithmetical results (iii) In Section 3, we present the application of the hyperbola parametrization to the study of lattice points on the Fermat equation (iv) In Section 4, In Section 4, we do a discussion on the likelihood of finding solutions to the Fermat factorization equation (v) In Section 5, we conclude
E authors declare that there are no conflicts of interest regarding the publication of this paper
Summary
One of fundamental research problems on conics is to find integral solutions of particular hyperbola parametrizations mainly x2 − y2 N over the integers, when N is a large semiprime, in which case if a computationally efficient algorithm is found, cryptosystems like RSA [11] would no longer be secured. We use the hyperbola parametrization introduced in [16] to study algebraic properties of lattice points and their distribution for Fermat’s factorization equation for which we find exact upper and lower bounds and we present the forms and cardinalities with a generalization of results for most of special cases of N, using results from the particular hyperbola parametrization. (i) In Section 1, we give an introduction (ii) In Section 2, we present the particular hyperbola parametrization and related arithmetical results (iii) In Section 3, we present the application of the hyperbola parametrization to the study of lattice points on the Fermat equation (iv) In Section 4, In Section 4, we do a discussion on the likelihood of finding solutions to the Fermat factorization equation (v) In Section 5, we conclude. BN(x, y)|Z (x, y) ∈ Z × Z/y2 x2 − 4Nx: braic set of all integral points on BN(x, y).
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