arXiv subjects
Publications and source records attributed to B. M. Cerna Maguiña.
In this paper we obtain bounds for integer solutions of quadratic polynomials in two variables that represent a natural number. Also we get some results on twin prime numbers. In addition, we use linear functionals to prove some results of the mathematical analysis and the Fermat's last theorem.
In this work we state a Theorem on number theory and apply it to solve some ordinary and partial differential equations.
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.