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

B. M. Cerna Maguiña; H. Blas; V. H. López Solís

paper

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

arXiv:1808.06145

Abstract

We consider a set of equations of the form $p_{j} (x, y) = (10 x + m_{j}) (10 y + n_{j}), x \geq 0, y \geq 0$ , $j = 1, 2, 3$ , 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 $\frac{p _{j}}{100} \leq A (p_{j}) B (p_{j}) \leq \frac{121}{1 0 ^{4}} p_{j}$ , 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.

10 pages, LaTex

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

Abstract

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