Some results on natural numbers represented by quadratic polynomials in two variables

We consider a set of equations of the form $p_j (x,y) = (10 x+m_j)(10 y + n_j),\,\,x\geq 0, y\geq0$, $j=1,2,3$, such that $\{m_1=7, n_1=3\}$, $\{m_2=n_2=9\}$ and $\{m_3=n_3=1\}$, respectively. It is shown that if $(a(p_j),b(p_j)) \in N \times N$ is a solution of the $j'$th equation one has the inequality $\frac{p_j}{100}\leq A(p_j) B(p_j) \leq \frac{121}{10^4} p_j$, where $A(p_j)\equiv a(p_j)+1, B(p_j)\equiv b(p_j)+1\,$ and $p_{j}$ is a natural number ending in 1, such that $\{A(p_1)\geq 4, B(p_1)\geq 8\}$, $\{A(p_2) \geq 2, B(p_2)\geq 2\}$, and $\{A(p_3) \geq 10, B(p_3)\geq 10\}$ hold, respectively. Moreover, assuming the previous result we show that $1\leq ( \frac{A(p_j+10) B(p_j+10)}{A(p_j) B(p_j)})^{1/100} \leq e^{0,000201} x (1+ \frac{10}{p_j})^{(0,101)^2}$, with $\{A(p_1)\geq 31, B(p_1)\geq 71\}$, $\{A(p_2) \geq 11, B(p_2)\geq 11\}$, and $\{A(p_3) \geq 91, B(p_3)\geq 91\}$, respectively. Finally, we present upper and lower bounds for the relevant positive integer solution of the equation defined by $p_j = (10 A+m_j)(10 B + n_j)$, for each case $j=1,2,3$, respectively.


Introduction
Number theory, and in particular the theory of prime numbers, still fascinates mathematicians and recently the physicists (see e.g. [7,8] and references therein). Nowadays, the data sets come from computer algorithms (see e.g. [6] for a highly optimized sieve of Eratosthenes implementation, which counts the primes below 10 10 in just 0.45 seconds), but mathematicians are still pursuing new patterns in primes. Recently, it has been shown that, contrary to the commom believe that the primes occur randomly, prime numbers have a peculiar dislike for other would-be primes which end in the same digit [4]. In fact, when performed a randomness check on the first 100 million primes, they found that a prime ending in 1 was followed by another prime ending in 1 only 18.5 percent of the time -different from the 25 percent you would expect given that primes greater than five can only end in one of four digits: 1, 3, 7, or 9. The current research focuses on finding new patterns in prime numbers (see e.g. [4,8] and references there in), as well as primes with largest number of digits (as of July 2018, the largest known prime is 2 77,232,917 − 1 with nearly 22 million digits long. It was found by the Great Internet Mersenne Prime Search (GIMPS) [5]).
Fermat-Euler prime number theorem states that a prime number p > 2 is a sum of two squares of integers, if and only if p ≡ 1(mod 4). In particular, m 2 + n 2 ∈ IP for infinitely many integers m, n. H. Iwaniec generalized [3] to polynomials of degree 2, in two variables, satisfying certain assumptions. Let P (m, n) = a m 2 +b mn+c n 2 +e m+f n+g be a primitive polynomial with integer cofficients. For P (m, n) reducible in Q[m, n] the Dirichlet's theorem on arithmetic pogression can be used to answer the question whether it represents infinitely many primes [2]. If P (m, n) is irreducible the theorem in [3] can be applied. In our previous contribution [1], we have discussed some aspects of the problem of obtaining a prime number starting from a given prime number.
In this paper, following our previous result on prime numbers we provide two new results, The second result assumes the above result as a hypothesis and establishes that 1 ≤ Moreover, we establish upper and lower bounds for the positive integer solutions of the equation p = (10x + 7)(10y + 3). Analogous results are obtained for the remaning cases ii) and iii).
In this work IN represents the set of natural numbers, e is the Euler's number, 2 Some natural numbers ending in 1 and quadratic polynomials. ii) p = (10a + 9)(10b + 9) iii) p = (10a + 1)(10b + 1), then, one has where A = a + 1, B = b + 1, with the next conditions for the relevant cases above Demonstration.
Let us consider the case i). So take p = (10x + 7)(10y + 3), with x ≥ 0, y ≥ 0. Next we So, for natural numbers X, Y one has shown that the equation (2.1) holds.
Observation 1. The demonstrations for the cases ii) and iii) follow analogous steps.
In addition, defining d ≡ 7x + 3y and form the eq. p = (10x + 7)(10y + 3) one can get . Analogous steps can be followed for the proofs in the cases ii) and iii).

Acknowledgments
BMCM would like to thank Concytec for partial financial support. HB would like to thank the members of the FC-UNASAM for hospitality and FC-UNI for partial financial support.