Abstract
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 is a solution of the j’th equation one has the inequality , where and pj is a natural number ending in 1, such that , and hold, respectively. Moreover, assuming the previous result we show that , with , and , 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.
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