mathematics

Schinzel's theorem

Schinzel's theorem 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 Schinzel's theorem rather than just read about it. In short: In the geometry of numbers, Schinzel's theorem is the following statement: It was originally proved by and named after Andrzej Schinzel. Proof Schinzel proved this theorem by the following construction.

Key takeaways

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

Reference excerpt

In the geometry of numbers, Schinzel's theorem is the following statement:

It was originally proved by and named after Andrzej Schinzel.

Proof

Schinzel proved this theorem by the following construction. If n {\displaystyle n} is an even number, with n = 2 k {\displaystyle n=2k} , then the circle given by the following equation passes through exactly n {\displaystyle n} points:

( x − 1 2 ) 2 + y 2 = 1 4 5 k − 1 . {\displaystyle \left(x-{\frac {1}{2}}\right)^{2}+y^{2}={\frac {1}{4}}5^{k-1}.}

This circle has radius 5 ( k − 1 ) / 2 / 2 {\displaystyle 5^{(k-1)/2}/2} , and is centered at the point ( 1 2 , 0 ) {\displaystyle ({\tfrac {1}{2}},0)} . For instance, the figure shows a circle with radius 5 / 2 {\displaystyle {\sqrt {5}}/2} through four integer points. Multiplying both sides of Schinzel's equation by four produces an equivalent equation in integers,

( 2 x − 1 ) 2 + ( 2 y ) 2 = 5 k − 1 . {\displaystyle \left(2x-1\right)^{2}+(2y)^{2}=5^{k-1}.}

This writes 5 k − 1 {\displaystyle 5^{k-1}} as a sum of two squares, where the first is odd and the second is even. There are exactly 4 k {\displaystyle 4k} ways to write 5 k − 1 {\displaystyle 5^{k-1}} as a sum of two squares, and half are in the order (odd, even) by symmetry. For example, 5 1 = ( ± 1 ) 2 + ( ± 2 ) 2 {\displaystyle 5^{1}=(\pm 1)^{2}+(\pm 2)^{2}} , so we have 2 x − 1 = 1 {\displaystyle 2x-1=1} or 2 x − 1 = − 1 {\displaystyle 2x-1=-1} , and 2 y = 2 {\displaystyle 2y=2} or 2 y = − 2 {\displaystyle 2y=-2} , which produces the four points pictured. On the other hand, if n {\displaystyle n} is odd, with n = 2 k + 1 {\displaystyle n=2k+1} , then the circle given by the following equation passes through exactly n {\displaystyle n} points:

( x − 1 3 ) 2 + y 2 = 1 9 5 2 k . {\displaystyle \left(x-{\frac {1}{3}}\right)^{2}+y^{2}={\frac {1}{9}}5^{2k}.}

This circle has radius 5 k / 3 {\displaystyle 5^{k}/3} , and is centered at the point ( 1 3 , 0 ) {\displaystyle ({\tfrac {1}{3}},0)} .

Properties The circles generated by Schinzel's construction are not the smallest possible circles passing through the given number of integer points, but they have the advantage that they are described by an explicit equation.

References

Worked examples

Example 1 — a first encounter with Schinzel's theorem

Start with the simplest possible case. Write down what Schinzel's theorem 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 Schinzel's theorem 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 Schinzel's theorem 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 Schinzel's theorem

In research: Schinzel's theorem 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 Schinzel's theorem 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: Schinzel's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometry of numbers, Lattice points, Theorems about circles, so understanding it makes those chapters shorter.
In everyday life: Look for Schinzel's theorem 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 Schinzel's theorem in 20 minutes

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

Frequently asked questions

What is Schinzel's theorem in simple terms?

In the geometry of numbers, Schinzel's theorem is the following statement: It was originally proved by and named after Andrzej Schinzel. Proof Schinzel proved this theorem by the following construction.

Why does Schinzel's theorem 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 Schinzel's theorem?

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 Schinzel's theorem.