number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, a Smith number is a

composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...

for which, in a given

number base In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...

, the sum of its digits is equal to the sum of the digits in its

prime factor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

ization in the given

. In the case of numbers that are not

square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ...

, 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 Lehigh University (LU) is a private research university in Bethlehem, Pennsylvania in the Lehigh Valley region of eastern Pennsylvania. The university was established in 1865 by businessman Asa Packer and was originally affiliated with the Epis ...

, as he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith: : 4937775 = 3¹ 5² 65837¹ while : 4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 · 1 + 5 · 2 + (6 + 5 + 8 + 3 + 7) · 1 = 42 in

base 10 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

.Sándor & Crstici (2004) p.383

Mathematical definition

Let

n

be a natural number. For base

b > 1

, let the function

F_(n)

be the

digit sum In mathematics, the digit sum of a natural number in a given number base is the sum of all its digits. For example, the digit sum of the decimal number 9045 would be 9 + 0 + 4 + 5 = 18. Definition Let n be a natural number. We define the digit ...

of n in base

b

. A natural number

n

has the integer factorisation :

n = \prod_ p^

and is a Smith number if :

F_b(n) = \sum_ v_p(n) F_b(p)

where

v_p(n)

is the

p-adic valuation In number theory, the valuation or -adic order of an integer is the exponent of the highest power of the prime number that divides . It is denoted \nu_p(n). Equivalently, \nu_p(n) is the exponent to which p appears in the prime factorization of ...

n

. For example, in

, 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

are: : 4, 22, 27, 58, 85, 94, 121, 166, 202,

265 __NOTOC__ Year 265 ( CCLXV) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Valerianus and Lucillus (or, less frequently, year 1018 ' ...

, 274, 319, 346,

355 __NOTOC__ Year 355 ( CCCLV) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Arbitio and Maesius (or, less frequently, year 1108 '' Ab ...

378 __NOTOC__ Year 378 ( CCCLXXVIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Valens and Augustus (or, less frequently, year 1131 ...

, 382, 391,

438 Year 438 (Roman numerals, CDXXXVIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Theodosius and Glabrio (or, less frequently, ye ...

, 454, 483, 517, 526,

535 __NOTOC__ Year 535 ( DXXXV) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Belisarius without colleague (or, less frequently, year 12 ...

, 562, 576, 588,

627 __NOTOC__ Year 627 ( DCXXVII) was a common year starting on Thursday of the Julian calendar. The denomination 627 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Eur ...

, 634,

636 Year 636 ( DCXXXVI) was a leap year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 636 for this year has been used since the early medieval period, when the Anno Domini calendar era became t ...

, 645, 648, 654,

663 __NOTOC__ Year 663 ( DCLXIII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. The denomination 663 for this year has been used since the early medieval period, when the Anno Domini calendar era ...

666 666 may refer to: * 666 (number) * 666 BC, a year * AD 666, a year * The number of the beast, a reference in the Book of Revelation in the New Testament Places * 666 Desdemona, a minor planet in the asteroid belt * U.S. Route 666, an America ...

690 __NOTOC__ Year 690 (Roman numerals, DCXC) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 690 for this year has been used since the early medieval period, when the Anno Domi ...

, 706,

728 __NOTOC__ Year 728 ( DCCXXVIII) was a leap year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 728 for this year has been used since the early medieval period, when the Anno Domini calendar e ...

729 Year 729 ( DCCXXIX) was a common year starting on Saturday of the Julian calendar. The denomination 729 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for nami ...

, 762,

778 __NOTOC__ Year 778 ( DCCLXXVIII) was a common year starting on Thursday of the Julian calendar. The denomination 778 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method ...

825 __NOTOC__ Year 825 ( DCCCXXV) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. Events By place India * A group of Persio-Assyrian adherents of the Church of the East, under the leade ...

, 852,

861 __NOTOC__ Year 861 ( DCCCLXI) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. Events By place Europe * March – Robert the Strong is appointed margrave of Neustria by King Ch ...

895 ' __NOTOC__ Year 895 (Roman numerals, DCCCXCV) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. Events By place Europe * The Hungarians, Magyars are expelled from southern Russia, and ...

, 913, 915,

922 __NOTOC__ Year 922 ( CMXXII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. Events By place Byzantine Empire * Summer – Battle of Constantinople: Emperor Romanos I sends Byza ...

958 Year 958 ( CMLVIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. Events By place Byzantine Empire * October / November – Battle of Raban: The Byzantines under John Tzimiskes ...

985 Year 985 ( CMLXXXV) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. Events By place Europe * Summer – Henry II (the Wrangler) is restored as duke of Bavaria by Empress Theoph ...

, 1086 …

Properties

W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers. The number of Smith numbers in

below 10^''n'' for ''n''=1,2,... is: : 1, 6, 49, 376, 3294, 29928, 278411, 2632758, 25154060, 241882509, … Two consecutive Smith numbers (for example, 728 and 729, or 2964 and 2965) are called Smith brothers.Sándor & Crstici (2004) p.384 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

for ''n'' = 1, 2, ... are: : 4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, … Smith numbers can be constructed from factored

repunit In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreat ...

s. The largest known Smith number in

is: :9 × R₁₀₃₁ × (10⁴⁵⁹⁴ + 3 + 1)¹⁴⁷⁶ where R₁₀₃₁ is a

equal to (10¹⁰³¹−1)/9.

Notes

References

* *

External links

* * Shyam Sunder Gupta
Fascinating Smith numbers
* {{Divisor classes Base-dependent integer sequences Lehigh University

Mathematical definition

Properties

See also

Notes

References

External links