Meertens number

Key takeaways

Meertens 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 Meertens number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Meertens number from memory before moving on to harder problems.

Reference excerpt

In number theory and mathematical logic, a Meertens number in a given number base b {\displaystyle b} is a natural number that is its own Gödel number. It was named after Lambert Meertens by Richard S. Bird as a present during the celebration of his 25 years at the CWI, Amsterdam.

Definition Let n {\displaystyle n} be a natural number. We define the Meertens function for base b > 1 {\displaystyle b>1} F b : N → N {\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} } to be the following:

F b ( n ) = ∏ i = 0 k − 1 p k − i − 1 d i . {\displaystyle F_{b}(n)=\prod _{i=0}^{k-1}p_{k-i-1}^{d_{i}}.}

where k = ⌊ log b ⁡ n ⌋ + 1 {\displaystyle k=\lfloor \log _{b}{n}\rfloor +1} is the number of digits in the number in base b {\displaystyle b} , p i {\displaystyle p_{i}} is the i {\displaystyle i} -th prime number (starting at 0), and

d i = n mod b i + 1 − n mod b i b i {\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}}

is the value of each digit of the number. A natural number n {\displaystyle n} is a Meertens number if it is a fixed point for F b {\displaystyle F_{b}} , which occurs if F b ( n ) = n {\displaystyle F_{b}(n)=n} . This corresponds to a Gödel encoding. For example, the number 3020 in base b = 4 {\displaystyle b=4} is a Meertens number, because

3020 = 2 3 3 0 5 2 7 0 {\displaystyle 3020=2^{3}3^{0}5^{2}7^{0}} . A natural number n {\displaystyle n} is a sociable Meertens number if it is a periodic point for F b {\displaystyle F_{b}} , where F b k ( n ) = n {\displaystyle F_{b}^{k}(n)=n} for a positive integer k {\displaystyle k} , and forms a cycle of period k {\displaystyle k} . A Meertens number is a sociable Meertens number with k = 1 {\displaystyle k=1} , and a amicable Meertens number is a sociable Meertens number with k = 2 {\displaystyle k=2} . The number of iterations i {\displaystyle i} needed for F b i ( n ) {\displaystyle F_{b}^{i}(n)} to reach a fixed point is the Meertens function's persistence of n {\displaystyle n} , and undefined if it never reaches a fixed point.

Meertens numbers and cycles of Fb for specific b All numbers are in base b {\displaystyle b} .

See also Arithmetic dynamics Dudeney number Factorion Happy number Kaprekar's constant Kaprekar number Narcissistic number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number

References

External links OEIS sequence A189398 (a(n) = 2^d(1) * 3^d(2) * ... * prime(k)^d(k)) OEIS sequence A246532 (Smallest Meertens number in base n, or -1 if none exists.)

Worked examples

Example 1 — a first encounter with Meertens number

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

In research: Meertens 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 Meertens 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: Meertens number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Arithmetic dynamics, Base-dependent integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Meertens 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.

How to study Meertens number in 20 minutes

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

Frequently asked questions

What is Meertens number in simple terms?

In number theory and mathematical logic, a Meertens number in a given number base b {\displaystyle b} is a natural number that is its own Gödel number. It was named after Lambert Meertens by Richard S.

Why does Meertens 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 Meertens 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 Meertens number.