Lattice Points on the Fermat Factorization Method
In this paper, we study algebraic properties of lattice points of the arc on the conics especially for , 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 , Fermat’s method fails. Otherwise, in terms of cardinality, it has, respectively, 4, 8, , , and lattice pointts if is an odd prime, with and being odd primes, with being prime, with being distinct primes, and with being odd primes. These results are important since they provide further arithmetical understanding and information on the integer solutions revealing factors of . These results could be particularly investigated for the purpose of improving the underlying integer factorization methods.
Diophantine equations have been for many decades a very important subject of research in number theory, and lattice points on curves have been studied in the literature particularly by Gauss, and bounds on arcs of conics have also been studied since then (see [1–6]). However, the necessity of representing an integer as difference of two squares, i.e., for a given , finding nontrivial couples such that , appears in the literature as the main idea of many factorization methods (see [7, 8]) as suggested by Fermat (see [9, 10]). While being not hard to observe, its lattice points are easily computable if one knows the factorization of , and in contrast, this gets exponentially harder when it comes to special cases of , mainly when , where are large primes, in which case this problem becomes equivalent to factoring the parameter .
For this reason, one of fundamental research problems on conics is to find integral solutions of particular hyperbola parametrizations mainly over the integers, particularly when is a large semiprime, in which case if a computationally efficient algorithm is found, cryptosystems like RSA  would no longer be secured.
Reviewing the literature, some results on various hyperbola parametrizations and their applications have been studied. Particularly in , Javier and Jorge used ideals in quadratic field to find an upper bound for the number of lattice points on Pell’s equation , while in , Jin et al. used results from the forms of integral solutions of the hyperbola to solve the same equation where d is of the form . In , the author studied a special case of hyperbola and presented the forms of its integral points over , and in , Yeonok investigated some behaviors of integral points on the hyperbola to the generalizations of Binet formula and Catalan’s identity, while in , the authors gave an application of group law on affine conics to cryptography. Still, in the previous works, algebraic properties and distribution of lattice points and cardinalities on Fermat’s equation are not presented. More recently, in , Gilda et al. investigated algebraic and arithmetical properties on the group structure , mainly isomorphisms, integral solutions, and a description of a factorization method with no generalization to the Fermat factorization equation.
In this paper, we use the hyperbola parametrization introduced in  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 , using results from the particular hyperbola parametrization.
The 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 finally conclude
Here is a list of the commonly used nomenclature in this paper: : algebraic set of all rational points on . : algebraic set of all integral points on . : algebraic set of integral points on whose coordinates are greater or equal to . : an injective homomorphism from to . : the cardinal of . : set of divisors of . : the set of all prime divisors of . : the Fermat factorization equation. : the Kronecker symbol.
2. BN Hyperbola Parametrization
A conic is an algebraic set satisfying an equation of the form where . Setting the parametrization defined by , in the projective space , we have .
At infinity, setting and considering , we obtain , and the equivalence class , and hence one of the points at infinity is .
From now on, , denotes over the field Q denotes over and denotes over the integers, i.e., ; and and .
Proposition 1. Consider the application is defined , , byThen, is an abelian group with neutral element .
Proof. Let us consider the affine space .
is a hyperbola of equation , where , i.e., , with . SetGiven two points ,We easily verify that .We have bothWe denote as the above defined additive law. This addition law is strongly unified since point doubling does exist and is well defined.
Now let us consider the applicationThis application is an internal composition law since , .(i)Associativity: given 3 points , here we show that .Note that this can be shown either geometrically or analytically, but here we give just the analytic proof.
Consider .We clearly see by identification that .(i)Neutral element: . It is obvious to see that . Given any point , from (3), we have Hence, .(ii)Symmetric element: is the symmetric element of P. It is obvious to see that . Then, we have Hence, .(iii)Commutativity:Hence, .
Proposition 2. Let and . Then, the following map:defines a group homomorphism.
Proof. , . . Thus, .
Consider and .Set the inverse of in ; by definition, .
Then, ., (it is the relation ). Then, relations , , and () imply that X is a group morphism
Furthermore, . is then injective and since such that . is also surjective. is bijective.
Since is a homomorphism of group and bijective, then defines an isomorphism of groups. Thus, and are isomorphic.
Definition 1. Given an integer , we define as the set of all divisors of . That is to say, .
Example 1. : ; .
Proposition 3. (i)If , such that , then , .(ii)If , with , primes, then and where , , .(iii)More generally, if .
Proof. (i)If , , . . Also, .(ii)If , , where , , . It comes to verify that verifies the equation . As , being prime and , then , , . We have . Hence, .(iii)More generally, if , .
Proposition 4. If , then the following holds:(i)