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

Cerna Maguiña B.M.; Blas H.; López Solís V.H. · Journal of Physics: Conference Series (2020) · DOI: 10.1088/1742-6596/1558/1/012011

Resumen

We consider a set of equations of the form Pj (x, y) = (10x + mj )(10y + nj ), x ≥ 0, y ≥ 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(pj),b(pj)) ϵ ℕ × ℕ is a solution of the j'th equation one has the inequality pj/100 ≤ A(pj)B(pj) ≤ 121/104 pj, where A(pj) ≡ a(pj)+1,B(pj) ≡ b(pj)+1 and pj is a natural number ending in 1, such that A(p1) ≥ 4,B(p1) ≥ 8}, {A(p2) ≥ 2,B(p2) ≥ 2}, and A(p3) ≥ 10,B(p3) ≥ 10} hold, respectively. Moreover, assuming the previous result we show that, with A(p1) ≥ 31,B(p1) ≥ 71}, A(p2) ≥ 11,B(p2) ≥ 11}, and A(p3) ≥ 91,B(p3) ≥ 91}, respectively. Finally, we present upper and lower bounds for the relevant positive integer solution of the equation defined by pj = (10A + mj )(10B + nj ), for each case j = 1, 2, 3, respectively. © Published under licence by IOP Publishing Ltd.

Instituciones: Universidad Nacional de Ingeniería; Universidad Nacional Santiago Antúnez de Mayolo