Pythagorean Triples

The investigation of Pythagorian triples has a very long history. For the first hundred years I refer to the famous book [DIC01]. Triangles of this type were given by Greek and Indian mathematicians. Arithmetically these are the solutions of the diophantine equation <math display='block'> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo><msup> <mi>y</mi> <mn>2</mn> </msup> <mo>=</mo><msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> </math> $${x^2} + {y^2} = {z^2}$$ in rational numbers. The general solution is given by the formulas <math display='block'> <math display='block'> <mrow> <mtable columnalign='left'> <mtr columnalign='left'> <mtd columnalign='left'> <mrow> <mi>x</mi><mo>=</mo><mi>l</mi><mo stretchy='false'>(</mo><msup> <mi>m</mi> <mn>2</mn> </msup> <mo>&#x2212;</mo><msup> <mi>n</mi> <mn>2</mn> </msup> <mo stretchy='false'>)</mo> </mrow> </mtd> </mtr> <mtr columnalign='left'> <mtd columnalign='left'> <mrow> <mi>y</mi><mo>=</mo><mi>l</mi><mo>&#x22C5;</mo><mn>2</mn><mi>m</mi><mi>n</mi> </mrow> </mtd> </mtr> <mtr columnalign='left'> <mtd columnalign='left'> <mrow> <mi>z</mi><mo>=</mo><mi>l</mi><mo stretchy='false'>(</mo><msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo><msup> <mi>n</mi> <mn>2</mn> </msup> <mo stretchy='false'>)</mo><mtext>&#x2009;</mtext><mi>l</mi><mo stretchy='false'>(</mo><mi>l</mi><mo>&#x2260;</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>,</mo><mtext>&#x2009;</mtext><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mtext>&#x2009;</mtext><mi>a</mi><mi>r</mi><mi>b</mi><mi>i</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>r</mi><mi>y</mi><mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math>$$\begin{array}{*{20}{l}} {x = l({m^2} - {n^2})} \\ {y = l \cdot 2mn} \\ {z = l({m^2} + {n^2})\;l(l \ne 0),\;m,n,\,arbitrary.} \end{array}$$.