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

Abstract

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.