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Siegel identity

Siegel identity is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Siegel identity rather than just read about it. In short: In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations. Statement The first formula is x 3 − x 1 x 2 − x 1 + x 2 − x 3 x 2 − x 1 = 1. {\displaystyle {\frac {x_{3}-x_{1}}{x_{2}-x_{1}}}+{\frac {x_{2}-x_{3}}{x_{2}-x_{1}}}=1.} The second is x 3 − x 1 x 2 − x 1 ⋅ t − x 2 t − x 3 + x 2 − x 3 x 2 − x 1 ⋅ t − x 1 t − x 3 = 1. {\displaystyle {\frac {x_{3}-x_{1…

Key takeaways

  • Siegel identity belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Siegel identity to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Siegel identity from memory before moving on to harder problems.

Reference excerpt

In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations.

Statement The first formula is

x 3 − x 1 x 2 − x 1 + x 2 − x 3 x 2 − x 1 = 1. {\displaystyle {\frac {x_{3}-x_{1}}{x_{2}-x_{1}}}+{\frac {x_{2}-x_{3}}{x_{2}-x_{1}}}=1.}

The second is

x 3 − x 1 x 2 − x 1 ⋅ t − x 2 t − x 3 + x 2 − x 3 x 2 − x 1 ⋅ t − x 1 t − x 3 = 1. {\displaystyle {\frac {x_{3}-x_{1}}{x_{2}-x_{1}}}\cdot {\frac {t-x_{2}}{t-x_{3}}}+{\frac {x_{2}-x_{3}}{x_{2}-x_{1}}}\cdot {\frac {t-x_{1}}{t-x_{3}}}=1.}

Application The identities are used in translating Diophantine problems connected with integral points on hyperelliptic curves into S-unit equations.

See also Siegel formula

References Baker, Alan (1975). Transcendental Number Theory. Cambridge University Press. p. 40. ISBN 0-521-20461-5. Zbl 0297.10013. Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. Vol. 9. Cambridge University Press. p. 53. ISBN 978-0-521-88268-2. Zbl 1145.11004. Kubert, Daniel S.; Lang, Serge (1981). Modular Units. Grundlehren der Mathematischen Wissenschaften. Vol. 244. ISBN 0-387-90517-0. Lang, Serge (1978). Elliptic Curves: Diophantine Analysis. Grundlehren der mathematischen Wissenschaften. Vol. 231. Springer-Verlag. ISBN 0-387-08489-4. Smart, N. P. (1998). The Algorithmic Resolution of Diophantine Equations. London Mathematical Society Student Texts. Vol. 41. Cambridge University Press. pp. 36–37. ISBN 0-521-64633-2.

Worked examples

Example 1 — a first encounter with Siegel identity

Start with the simplest possible case. Write down what Siegel identity claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Siegel identity before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Siegel identity ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Siegel identity

In research
Siegel identity appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Siegel identity in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Siegel identity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic identities, Diophantine equations, so understanding it makes those chapters shorter.
In everyday life
Look for Siegel identity outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Siegel identity in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Siegel identity means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Siegel identity out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Siegel identity in simple terms?

In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations. Statement The first formula is x 3 − x 1 x 2 − x 1 + x 2 − x 3 x 2 − x 1 = 1. {\displaystyle {\frac {x_{3}-x_{1}}{x_{2}-x_{1}}}+{\frac {x_{2}-x_{3}}{x_{2}-x_{1}}}=1.} The second…

Why does Siegel identity matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Siegel identity?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Siegel identity.

Tags

  • Algebraic identities
  • Diophantine equations

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