The Pythagorean theorem <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${x}^{2}+{y}^{2}={z}^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <msup> <mrow> <mi>z</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is usually stated over the integers <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Z$</EquationSource> <EquationSource Format="MATHML"><math> <mi>Z</mi> </math></EquationSource> </InlineEquation>, a subring of the reals <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R$</EquationSource> <EquationSource Format="MATHML"><math> <mi>R</mi> </math></EquationSource> </InlineEquation>, and it holds true for infinitely many solutions. We explore the theorem over subrings of other number spaces, such as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H$</EquationSource> <EquationSource Format="MATHML"><math> <mi>H</mi> </math></EquationSource> </InlineEquation> (quaternions) and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$O$</EquationSource> <EquationSource Format="MATHML"><math> <mi>O</mi> </math></EquationSource> </InlineEquation> (octonions). We present several results mainly for L (Lipschitz quaternions) and provide a geometric interpretation of the theorem in the subring <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>. Some results for the corresponding subring G of O are also presented. Finally, we also present some results for the rings <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$H/{Z}_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo stretchy="false">/</mo> <msub> <mi>Z</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$O/{Z}_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">/</mo> <msub> <mi>Z</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>.

The pythagorean theorem in quaternion and octonion spaces

期刊论文
发表日期：

2026年5月25日

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Michael Aristidou,
George Chailos

摘要

The Pythagorean theorem ${x}^{2}+{y}^{2}={z}^{2}$ $x^{2} + y^{2} = z^{2}$ is usually stated over the integers $Z$ $Z$ , a subring of the reals $R$ $R$ , and it holds true for infinitely many solutions. We explore the theorem over subrings of other number spaces, such as $H$ $H$ (quaternions) and $O$ $O$ (octonions). We present several results mainly for L (Lipschitz quaternions) and provide a geometric interpretation of the theorem in the subring $L$ $L$ . Some results for the corresponding subring G of O are also presented. Finally, we also present some results for the rings $H/{Z}_{p}$ $H / Z_{p}$ and $O/{Z}_{p}$ $O / Z_{p}$ .