Repdigit

Key takeaways

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

Reference excerpt

In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit". Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which are repdigits when represented in binary). Any such number can be represented as follows

n n … n n ⏟ k = ( n n − n ) k − n k ( n n − 2 ⋅ n ) ⋅ n ( k − 2 ) {\displaystyle \underbrace {nn\ldots nn} _{k}={\frac {(nn-n)^{k}-n^{k}}{(nn-2\cdot n)\cdot n^{(k-2)}}}}

Where nn is the concatenation of n with n. k the number of concatenated n. nn can be represented mathematically as

n ⋅ ( 10 ⌊ log 10 ⁡ ( n ) ⌋ + 1 + 1 ) {\displaystyle n\cdot \left(10^{\lfloor \log _{10}(n)\rfloor +1}+1\right)}

for n = 23 and k = 5, the formula will look like this

( 2323 − 23 ) 5 − 23 5 ( 2323 − 2 ⋅ 23 ) ⋅ 23 ( 5 − 2 ) = 64363429993563657 27704259 = 2323232323 ⏟ 5 {\displaystyle {\frac {(2323-23)^{5}-23^{5}}{(2323-2\cdot 23)\cdot 23^{(5-2)}}}={\frac {64363429993563657}{27704259}}=\underbrace {2323232323} _{5}}

However, 2323232323 is not a repdigit. Also, any number can be decomposed into the sum and difference of the repdigit numbers. For example 3453455634 = 3333333333 + (111111111 + (9999999 - (999999 - (11111 + (77 + (2)))))) Repdigits are the representation in base B {\displaystyle B} of the number x B y − 1 B − 1 {\displaystyle x{\frac {B^{y}-1}{B-1}}} where 0 < x < B {\displaystyle 0<x<B} is the repeated digit and 1 < y {\displaystyle 1<y} is the number of repetitions. For example, the repdigit 77777 in base 10 is 7 × 10 5 − 1 10 − 1 {\displaystyle 7\times {\frac {10^{5}-1}{10-1}}} . A variation of repdigits called Brazilian numbers are numbers that can be written as a repdigit in some base, not allowing the repdigit 11, and not allowing the single-digit numbers (or all numbers will be Brazilian). For example, 27 is a Brazilian number because 27 is the repdigit 33 in base 8, while 9 is not a Brazilian number because its only repdigit representation is 118, not allowed in the definition of Brazilian numbers. The representations of the form 11 are considered trivial and are disallowed in the definition of Brazilian numbers, because all natural numbers n greater than two have the representation 11n − 1. The first twenty Brazilian numbers are

7, 8, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 33, ... (sequence A125134 in the OEIS). On some websites (including imageboards like 4chan), it is considered an auspicious event when the sequentially-assigned ID number of a post is a repdigit, such as 22,222,222, which is one type of "GET" (others including round numbers like 34,000,000, or sequential digits like 12,345,678).

History The concept of a repdigit has been studied under that name since at least 1974; earlier Beiler (1966) called them "monodigit numbers". The Brazilian numbers were introduced later, in 1994, in the 9th Iberoamerican Mathematical Olympiad that took place in Fortaleza, Brazil. The first problem in this competition, proposed by Mexico, was as follows:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Repdigit

Start with the simplest possible case. Write down what Repdigit claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Repdigit 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 Repdigit 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 Repdigit

In research: Repdigit appears in science 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 Repdigit 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: Repdigit is common in secondary-school and first-year university syllabi. It links to neighbouring topics Base-dependent integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Repdigit 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 Repdigit in 20 minutes

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

Frequently asked questions

What is Repdigit in simple terms?

In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit".

Why does Repdigit matter?

Because it connects several science 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 Repdigit?

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 Repdigit.