Smith number

Key takeaways

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

Reference excerpt

In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as many times as needed. Smith numbers were named by Albert Wilansky of Lehigh University, as he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:

4937775 = 3 · 5 · 5 · 65837 while

4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + (6 + 5 + 8 + 3 + 7) in base 10.

Mathematical definition Let n {\displaystyle n} be a natural number. For base b > 1 {\displaystyle b>1} , let the function F b ( n ) {\displaystyle F_{b}(n)} be the digit sum of n {\displaystyle n} in base b {\displaystyle b} . A natural number n {\displaystyle n} with prime factorization

n = ∏ p prime p ∣ n , p v p ( n ) {\displaystyle n=\prod _{\stackrel {p\mid n,}{p{\text{ prime}}}}p^{v_{p}(n)}}

is a Smith number if

F b ( n ) = ∑ p prime p ∣ n , v p ( n ) F b ( p ) . {\displaystyle F_{b}(n)=\sum _{\stackrel {p\mid n,}{p{\text{ prime}}}}v_{p}(n)F_{b}(p).}

Here the exponent v p ( n ) {\displaystyle v_{p}(n)} is the multiplicity of p {\displaystyle p} as a prime factor of n {\displaystyle n} (also known as the p-adic valuation of n {\displaystyle n} ). For example, in base 10, 378 = 21 · 33 · 71 is a Smith number since 3 + 7 + 8 = 2 · 1 + 3 · 3 + 7 · 1, and 22 = 21 · 111 is a Smith number, because 2 + 2 = 2 · 1 + (1 + 1) · 1. The first few Smith numbers in base 10 are

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985. (sequence A006753 in the OEIS)

Properties W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers. The number of Smith numbers in base 10 below 10n for n = 1, 2, ... is given by

1, 6, 49, 376, 3294, 29928, 278411, 2632758, 25154060, 241882509, ... (sequence A104170 in the OEIS). Two consecutive Smith numbers (for example, 728 and 729, or 2964 and 2965) are called Smith brothers. It is not known how many Smith brothers there are. The starting elements of the smallest Smith n-tuple (meaning n consecutive Smith numbers) in base 10 for n = 1, 2, ... are

4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, ... (sequence A059754 in the OEIS). Smith numbers can be constructed from factored repunits. As of 2010, the largest known Smith number in base 10 is

9 × R1031 × (104594 + 3×102297 + 1)1476 ×103913210 where R1031 is the base 10 repunit (101031 − 1)/9.

Worked examples

Example 1 — a first encounter with Smith number

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

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

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

Frequently asked questions

What is Smith number in simple terms?

In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the…

Why does Smith 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 Smith 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 Smith number.