Lattice Points on the Fermat Factorization Method

Abstract

In this paper, we study algebraic properties of lattice points of the arc on the conics $x^2-d{y}^2=N$ 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 $\left(N-6\right)\equiv 0\mathrm{mod}4$ , Fermat’s method fails. Otherwise, in terms of cardinality, it has, respectively, 4, 8, $2\left(\alpha +1\right)$ , $\left(1-{\delta }_{2p_i}\right)2^{n+1}$ , and $2{\prod }_{i=1}^n\left({\alpha }_i+1\right)$ lattice pointts if $N$ is an odd prime, $N=N_a\times N_b$ with $N_a$ and $N_b$ being odd primes, $N=N_a^{\alpha }$ with $N_a$ being prime, $N={\prod }_{i=1}^n{p}_i$ with $p_i$ being distinct primes, and $N={\prod }_{i=1}^n{N}_i^{{\alpha }_i}$ with $N_i$ 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.