A New Proof for Congruent Number’s Problem via Pythagorician Divisors
- 1 UFR Mathematics and Computer Science, Félix Houphouët Boigny University, Abidjan, Cote d’Ivoire
- 2 UFR Mathematics and Computer Science, Félix Houphouët Boigny University, Abidjan, Cote d’Ivoire
Abstract
Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mrow><mo>(</mo> <mrow> <mi>a</mi><mn>,</mn><mi>b</mi><mn>,</mn><mi>c</mi></mrow> <mo>)</mo></mrow><mo>∈</mo><msup> <mi>ℕ</mi> <mrow> <mn>3</mn><mo>∗</mo></mrow> </msup> </mrow> </math> , we give a new proof of the well-known problem of these particular squareless numbers <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>n</mi><mo>∈</mo><msup> <mi>ℕ</mi> <mo>∗</mo> </msup> </mrow> </math> , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msup> <mrow> <mrow><mo>(</mo> <mrow> <mfrac> <mi>A</mi> <mi>α</mi> </mfrac> </mrow> <mo>)</mo></mrow></mrow> <mn>2</mn> </msup> <mo>+</mo><msup> <mrow> <mrow><mo>(</mo> <mrow> <mfrac> <mi>B</mi> <mi>β</mi> </mfrac> </mrow> <mo>)</mo></mrow></mrow> <mn>2</mn> </msup> <mo>=</mo><msup> <mrow> <mrow><mo>(</mo> <mrow> <mfrac> <mi>C</mi> <mi>γ</mi> </mfrac> </mrow> <mo>)</mo></mrow></mrow> <mn>2</mn> </msup> </mrow> </math> , such that its area <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>Δ</mi><mo>=</mo><mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mfrac> <mi>A</mi> <mi>α</mi> </mfrac> <mfrac> <mi>B</mi> <mi>β</mi> </mfrac> <mo>=</mo><mi>n</mi></mrow> </math> ; or in an equivalent way, to that of the existence of numbers <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msup> <mi>U</mi> <mn>2</mn> </msup> <mn>,</mn><msup> <mi>V</mi> <mn>2</mn> </msup> <mn>,</mn><msup> <mi>W</mi> <mn>2</mn> </msup> <mo>∈</mo><msup> <mi>ℚ</mi> <mrow> <mn>2</mn><mo>∗</mo></mrow> </msup> </mrow> </math> that are in an arithmetic progression of reason <i>n</i>; Problem equivalent to the existence of: <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mrow><mo>(</mo> <mrow> <mi>a</mi><mn>,</mn><mi>b</mi><mn>,</mn><mi>c</mi></mrow> <mo>)</mo></mrow><mo>∈</mo><msup> <mi>ℕ</mi> <mrow> <mn>3</mn><mo>∗</mo></mrow> </msup> </mrow> </math> prime in pairs, and <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>f</mi><mo>∈</mo><msup> <mi>ℕ</mi> <mo>∗</mo> </msup> </mrow> </math> , such that: <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msup> <mrow> <mrow><mo>(</mo> <mrow> <mfrac> <mrow> <mi>a</mi><mo>−</mo><mi>b</mi></mrow> <mrow> <mn>2</mn><mi>f</mi></mrow> </mfrac> </mrow> <mo>)</mo></mrow></mrow> <mn>2</mn> </msup> </mrow> </math> , <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msup> <mrow> <mrow><mo>(</mo> <mrow> <mfrac> <mi>c</mi> <mrow> <mn>2</mn><mi>f</mi></mrow> </mfrac> </mrow> <mo>)</mo></mrow></mrow> <mn>2</mn> </msup> </mrow> </math> , <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msup> <mrow> <mrow><mo>(</mo> <mrow> <mfrac> <mrow> <mi>a</mi><mo>+</mo><mi>b</mi></mrow> <mrow> <mn>2</mn><mi>f</mi></mrow> </mfrac> </mrow> <mo>)</mo></mrow></mrow> <mn>2</mn> </msup> </mrow> </math> are in an arithmetic progression of reason <i>n</i> ; And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo><msup> <mi>b</mi> <mn>2</mn> </msup> <mo>=</mo><msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> </math> , such that its area <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>Δ</mi><mo>=</mo><mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>a</mi><mi>b</mi><mo>=</mo><mi>n</mi><msup> <mi>f</mi> <mn>2</mn> </msup> </mrow> </math> , where <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>f</mi><mo>∈</mo><msup> <mi>ℕ</mi> <mo>∗</mo> </msup> </mrow> </math> , and this last equation can be written as follows, when using Pythagorician divisors: (1) <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>Δ</mi><mo>=</mo><mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>a</mi><mi>b</mi><mo>=</mo><msup> <mn>2</mn> <mrow> <mi>S</mi><mo>−</mo><mn>1</mn></mrow> </msup> <mi>d</mi><mover accent='true'> <mi>e</mi> <mo>¯</mo> </mover> <mrow><mo>(</mo> <mrow> <mi>d</mi><mo>+</mo><msup> <mn>2</mn> <mrow> <mi>S</mi><mo>−</mo><mn>1</mn></mrow> </msup> <mover accent='true'> <mi>e</mi> <mo>¯</mo> </mover> </mrow> <mo>)</mo></mrow><mrow><mo>(</mo> <mrow> <mi>d</mi><mo>+</mo><msup> <mn>2</mn> <mi>S</mi> </msup> <mover accent='true'> <mi>e</mi> <mo>¯</mo> </mover> </mrow> <mo>)</mo></mrow><mo>=</mo><mi>n</mi><msup> <mi>f</mi> <mn>2</mn> </msup> <mn><mo>;</mo></mn></mrow> </math> Where <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mrow><mo>(</mo> <mrow> <mi>d</mi><mn>,</mn><mover accent='true'> <mi>e</mi> <mo>¯</mo> </mover> </mrow> <mo>)</mo></mrow><mo>∈</mo><msup> <mrow> <mrow><mo>(</mo> <mrow> <mn>2</mn><mi>ℕ</mi><mo>+</mo><mn>1</mn></mrow> <mo>)</mo></mrow></mrow> <mn>2</mn> </msup> </mrow> </math> such that <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>g</mi><mi>c</mi><mi>d</mi><mrow><mo>(</mo> <mrow> <mi>d</mi><mn>,</mn><mover accent='true'> <mi>e</mi> <mo>¯</mo> </mover> </mrow> <mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow> </math> and <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>S</mi><mo>∈</mo><msup> <mi>ℕ</mi> <mo>∗</mo> </msup> </mrow> </math> , where <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msup> <mn>2</mn> <mrow> <mi>S</mi><mo>−</mo><mn>1</mn></mrow> </msup> </mrow> </math> , <i>d</i>, <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mover accent='true'> <mi>e</mi> <mo>¯</mo> </mover> </math> , <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>d</mi><mo>+</mo><msup> <mn>2</mn> <mrow> <mi>S</mi><mo>−</mo><mn>1</mn></mrow> </msup> <mover accent='true'> <mi>e</mi> <mo>¯</mo> </mover> </mrow> </math> , <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>d</mi><mo>+</mo><msup> <mn>2</mn> <mi>S</mi> </msup> <mover accent='true'> <mi>e</mi> <mo>¯</mo> </mover> </mrow> </math> , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>n</mi><mo>=</mo><mn>1</mn></mrow> </math> , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>n</mi><mo>=</mo><mn>1</mn></mrow> </math> (resp. <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>n</mi><mo>=</mo><mn>2</mn></mrow> </math> ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>n</mi><mo>=</mo><mi>p</mi><mo>≡</mo><mi>a</mi></mrow> </math> (<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mrow><mo>(</mo> <mrow> <mi>mod</mi><mn>8</mn></mrow> <mo>)</mo></mrow></mrow> </math> , <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>g</mi><mi>c</mi><mi>d</mi><mrow><mo>(</mo> <mrow> <mi>a</mi><mn>,8</mn></mrow> <mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow> </math> ). To prove it, we use a classical proof by induction on <i>k</i> , that shows the non-solvability property of any of the following systems (<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>t</mi><mo>=</mo><mn>0</mn></mrow> </math> , corresponding to case <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>n</mi><mo>=</mo><mn>1</mn></mrow> </math> (resp. <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>t</mi><mo>=</mo><mn>1</mn></mrow> </math> , corresponding to case <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>n</mi><mo>=</mo><mn>2</mn></mrow> </math> )): <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mrow><mo>(</mo> <mrow> <msub> <mi>Ξ</mi> <mrow> <mi>t</mi><mo>,</mo><mi>k</mi></mrow> </msub> </mrow> <mo>)</mo></mrow><mrow><mo>{</mo> <mrow> <mtable columnalign='left'> <mtr columnalign='left'> <mtd columnalign='left'> <mrow> <msup> <mi>X</mi> <mn>2</mn> </msup> <mo>+</mo><msup> <mn>2</mn> <mi>t</mi> </msup> <msup> <mrow> <mrow><mo>(</mo> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> <mi>Y</mi></mrow> <mo>)</mo></mrow></mrow> <mn>2</mn> </msup> <mo>=</mo><msup> <mi>Z</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr columnalign='left'> <mtd columnalign='left'> <mrow> <msup> <mi>X</mi> <mn>2</mn> </msup> <mo>+</mo><msup> <mn>2</mn> <mrow> <mi>t</mi><mo>+</mo><mn>1</mn></mrow> </msup> <msup> <mrow> <mrow><mo>(</mo> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> <mi>Y</mi></mrow> <mo>)</mo></mrow></mrow> <mn>2</mn> </msup> <mo>=</mo><msup> <mi>T</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable></mrow> </mrow></mrow> </math> , where <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow> </math> ; and solutions <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mrow><mo>(</mo> <mrow> <mi>X</mi><mn>,</mn><mi>Y</mi><mn>,</mn><mi>Z</mi><mn>,</mn><mi>T</mi></mrow> <mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo> <mrow> <msub> <mi>D</mi> <mi>k</mi> </msub> <mn>,</mn><msub> <mi>E</mi> <mi>k</mi> </msub> <mo>,</mo><msub> <mi>f</mi> <mi>k</mi> </msub> <mo>,</mo><msub> <msup> <mi>f</mi> <mo>′</mo> </msup> <mi>k</mi> </msub> </mrow> <mo>)</mo></mrow><mo>∈</mo><msup> <mrow> <mrow><mo>(</mo> <mrow> <mn>2</mn><mi>ℕ</mi><mo>+</mo><mn>1</mn></mrow> <mo>)</mo></mrow></mrow> <mn>4</mn> </msup> </mrow> </math> , are given in pairwise prime numbers.<br /><b>2020-Mathematics Subject Classification </b><br />11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25
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