mathematics

Størmer number

Størmer number 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 Størmer number rather than just read about it. In short: In mathematics, a Størmer number or arc-cotangent irreducible number is a positive integer n {\displaystyle n} for which the greatest prime factor of n 2 + 1 {\displaystyle n^{2}+1} is greater than or equal to 2 n {\displaystyle 2n} . They are named after Carl Størmer.

Key takeaways

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

Reference excerpt

In mathematics, a Størmer number or arc-cotangent irreducible number is a positive integer n {\displaystyle n} for which the greatest prime factor of n 2 + 1 {\displaystyle n^{2}+1} is greater than or equal to 2 n {\displaystyle 2n} . They are named after Carl Størmer.

Sequence The first Størmer numbers below 100 are:

The complementary sequence (numbers below 100 that aren't Størmer) is only 3, 7, 8, 13, 17, 18, 21, 30, 31, 32, 38, 41, 43, 46, 47, 50, 55, 57, 68, 70, 72, 73, 75, 76, 83, 91, 93, 98, 99 and 100.

Density John Todd proved that this sequence is neither finite nor cofinite.

More precisely, the natural density of the Størmer numbers lies between 0.5324 and 0.905. It has been conjectured that their natural density is the natural logarithm of 2, approximately 0.693, but this remains unproven. Because the Størmer numbers have positive density, the Størmer numbers form a large set.

Application The Størmer numbers arise in connection with the problem of representing the Gregory numbers (arctangents of rational numbers) G a / b = arctan ⁡ b a {\displaystyle G_{a/b}=\arctan {\frac {b}{a}}} as sums of Gregory numbers for integers (arctangents of unit fractions). The Gregory number G a / b {\displaystyle G_{a/b}} may be decomposed by repeatedly multiplying the Gaussian integer a + b i {\displaystyle a+bi} by numbers of the form n ± i {\displaystyle n\pm i} , in order to cancel prime factors p {\displaystyle p} from the imaginary part; here n {\displaystyle n} is chosen to be a Størmer number such that n 2 + 1 {\displaystyle n^{2}+1} is divisible by p {\displaystyle p} .

References

Worked examples

Example 1 — a first encounter with Størmer number

Start with the simplest possible case. Write down what Størmer number 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 Størmer number 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 Størmer number 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 Størmer number

In research: Størmer number 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 Størmer number 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: Størmer number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Størmer number 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 Størmer number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Størmer number 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 Størmer number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Størmer number in simple terms?

Why does Størmer number 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 Størmer number?

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 Størmer number.