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<p>Let <i>S</i> = {1, 10, <i>c</i>} be an integer set. In this paper, we show that if the product of any two elements of <i>S</i>, decreased by one, forms a perfect square, and if there exists a positive integer <i>d</i> such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(d+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(10d+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>10</mn> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(cd+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> are all perfect squares, then <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(d=d^{\pm}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <msup> <mi>d</mi> <mo>±</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> where<Equation ID="Equ1"> <EquationSource Format="TEX">\(d^{\pm}=19c-11\pm 6\sqrt{(c-1)(10c-1)}.\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mi>d</mi> <mo>±</mo> </msup> <mo>=</mo> <mn>19</mn> <mi>c</mi> <mo>-</mo> <mn>11</mn> <mo>±</mo> <mn>6</mn> <msqrt> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mi>c</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msqrt> <mo>.</mo> </mrow> </math></EquationSource> </Equation>Additionally, under certain conditions, we investigate the integer points on the parametrized family of elliptic curves:<Equation ID="Equ2"> <EquationSource Format="TEX">\(E: y^2 = (x+1)(10x+1)(cx+1).\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>E</mi> <mo>:</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>10</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation></p>

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On the D(1) extension of the \(D(-1)\)-Diophantine triples induced by the \(D(-1)\)-Diophantine pair {1,10}

  • 期刊论文

  • 发表日期：

    2025年10月16日

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  • Y. Briedj,
  • S. E. Rihane

摘要

Let S = {1, 10, c} be an integer set. In this paper, we show that if the product of any two elements of S, decreased by one, forms a perfect square, and if there exists a positive integer d such that \(d+1\) d + 1 , \(10d+1\) 10 d + 1 and \(cd+1\) c d + 1 are all perfect squares, then \(d=d^{\pm}\) d = d ± where \(d^{\pm}=19c-11\pm 6\sqrt{(c-1)(10c-1)}.\) d ± = 19 c - 11 ± 6 ( c - 1 ) ( 10 c - 1 ) . Additionally, under certain conditions, we investigate the integer points on the parametrized family of elliptic curves: \(E: y^2 = (x+1)(10x+1)(cx+1).\) E : y 2 = ( x + 1 ) ( 10 x + 1 ) ( c x + 1 ) .