mathematics

Smarandache–Wellin number

Smarandache–Wellin 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 Smarandache–Wellin number rather than just read about it. In short: In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R.

Key takeaways

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

Reference excerpt

In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R. Wellin. The first decimal Smarandache–Wellin numbers are:

2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, 2357111317192329, ... (sequence A019518 in the OEIS).

Smarandache–Wellin prime A Smarandache–Wellin number that is also prime is called a Smarandache–Wellin prime. The first three are 2, 23 and 2357 (sequence A069151 in the OEIS). The fourth is 355 digits long: it is the result of concatenating the first 128 prime numbers, through 719. The primes at the end of the concatenation in the Smarandache–Wellin primes are

2, 3, 7, 719, 1033, 2297, 3037, 11927, ... (sequence A046284 in the OEIS). The indices of the Smarandache–Wellin primes in the sequence of Smarandache–Wellin numbers are:

1, 2, 4, 128, 174, 342, 435, 1429, ... (sequence A046035 in the OEIS). The 1429th Smarandache–Wellin number is a prime with 5719 digits ending in 11927, discovered by Eric W. Weisstein as a probable prime in 1998 and then proven prime in 2022. In March 2009, Weisstein's search showed the index of the next Smarandache–Wellin prime (if one exists) is at least 22077.

See also Copeland–Erdős constant Champernowne constant, another example of a number obtained by concatenating a representation in a given base.

References

External links Weisstein, Eric W. "Smarandache–Wellin number". MathWorld. Weisstein, Eric W. "Smarandache–Wellin prime". MathWorld. "Smarandache-Wellin number". PlanetMath. List of first 54 Smarandache–Wellin numbers with factorizations Smarandache–Wellin primes at The Prime Glossary Smith, S. "A Set of Conjectures on Smarandache Sequences." Bull. Pure Appl. Sci. 15E, 101–107, 1996.

Worked examples

Example 1 — a first encounter with Smarandache–Wellin number

Start with the simplest possible case. Write down what Smarandache–Wellin 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 Smarandache–Wellin 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 Smarandache–Wellin 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 Smarandache–Wellin number

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

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Smarandache–Wellin number” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Smarandache–Wellin number in 20 minutes

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

Frequently asked questions

What is Smarandache–Wellin number in simple terms?

Why does Smarandache–Wellin 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 Smarandache–Wellin 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 Smarandache–Wellin number.