Some Results on Integer Solutions of Quadratic Polynomials in Two Variables

Although it is true that there are several articles that study quadratic equations in two variables, they do so in a general way. We focus on the study of natural numbers ending in one, because the other cases can be studied in a similar way. We have given the subject a different approach, that is why our bibliographic citations are few. In this work, using basic tools of functional analysis, we achieve some results in the study of integer solutions of quadratic polynomials in two variables that represent a given natural number. To determine if a natural number ending in one is prime, we must solve equations (i) P = (10 x + 9)(10 y + 9) , (ii) P = (10 x + 1)(10 y + 1) , (iii) P = (10 x + 7)(10 y + 3) . If these equations do not have an integer solution, then the number P is prime. The advantage of this technique is that, to determine if a natural number p is prime, it is not necessary to know the prime numbers less than or equal to the square root of p. The objective of this work was to reduce the number of possibilities assumed by the integer variables ( x, y ) in the equation (i) , (ii) , (iii) respectively. Although it is true that this objective was achieved, we believe that the lower limits for the sums of the solutions of equations (i) , (ii) , (iii) , were not optimal, since in our recent research we have managed to obtain limits lower, which reduce the domain of the integer variables ( x, y ) solve equations (i) , (ii) , (iii) , respectively. In a future article we will show the results obtained. The methodology used was deductive and inductive. We would have liked to have a supercomputer, to build or determine prime numbers of many millions of digits, but this is not possible, since we do not have the support of our respective authorities. We believe that the contribution of this work to number theory is the creation of linear functionals for the study of integer solutions of quadratic polynomials in two variables, which represent a given natural number. The utility of large prime numbers can be used to encode any type of information safely, and the scheme shown in this article could be useful for this process.


Introduction
It is a non trivial task to determine whether a number is prime or not. The question if a number is prime or not has always attracted mathematicians and number theorists who have obtained some partial successes; but no one has yet obtained an exact mathematical result and acceptable algorithmic structure that solves this question for all or an indefinite set of prime numbers.
In our previous contribution [1], we have discussed quadratic polynomials in two variables that represent natural numbers N that end in 1. In this work we improve the results of the article [1], in such a way that in the present contribution we have decreased (in computational terms) the number of steps to determine the entire solution of the quadratic polynomial that represents the natural number N that ends in 1.
We know that to determine the primality of N it is enough to prove that the number is not divisible by prime numbers less than √ N . Wilson's Theorem [2] allows us to determine exactly when a number is prime: N ∈ N is prime if and only if (N − 1)! + 1 is multiple of N . The problem of Wilson's method comes when we have larger numbers since the factorial becomes very large, so it is definitely not a practical method.
In this work we use elementary methods to offer alternative methods to determine the primality of any natural number.
then from (1) and (2) we have As λ 2 ∈ Q, let λ 2 = m n ; m, n relatively prime numbers, then we have the following quadratic polynomial defining q = 10R one has that the general solution is Using Fermat's last theorem [3] for case 2 there is k ∈ Z such that Denoting k = M 2 − N 2 , Rm = 2M N , n = M 2 + N 2 and replacing these relationships in (4), we have Replacing (5) in (2), we have Since (x 0 , y 0 ) ∈ R 2 , for x 0 = y 0 = 1, we have 100AB + 90(A + B) = p − 81 which is equivalent to (10A + 9)(10B + 9) = p; so, p has been factorized. In Example 2.1. Consider the number p = 900071. So, taking into account the above relationships one has For A ≥ 11 and A < B from the Theorem 2 one gets (see also [1]) Let us define τ as From the equations (6), (7) and (8) one gets In addition, from (6) and (8) one can get The last inequalities possess a solution provided that Using (10) and where N and M are relatively prime numbers, with Proof. From the theorem (2.1) we have From this relationship, the equation (1) and the Pythagoras' theorem we have This last relation will be used to obtain the result we are searching for. Next, from the relation  , In order to motivate the theorems below some comments are in order here. Notice that for a p ∈ N ending in 1 the quadratic equation in two variables which represents such a number can be somewhat approximated by another quadratic equation which represents a natural number p + 101. This number can be factorized easily since 1 is a small natural number. The factors of p + 101 would be used as upper and lower bounds in order to get the solutions of the quadratic equation representing the number p.
Proof. The proof is presented only for the case It is important to emphasize that this is not the only representation of p + 10 but the representation considered will allow us to obtain the desired upper and lower bounds. Then we consider C = 0, since for this case on has p + 10 = Therefore from (18) and (19) we obtain the desired inequality.
We would like p + 10L to be a multiple of Therefore, from (27) and (28), we obtain the desired inequality. Similarly we can proceed with the other values of L.