Smith number
Encyclopedia
A Smith number is a composite number
for which, in a given base
(in base 10
by default), the sum of its digits
is equal to the sum of the digits in its prime factor
ization. For example, 378 = 2 × 3 × 3 × 3 × 7 is a Smith number since 3 + 7 + 8 = 2 + 3 + 3 + 3 + 7. It's important to remember that, by definition, the factors are treated as digits. For example, 22 factors to 2 × 11 and yields three digits: 2, 1, 1. Therefore 22 is a Smith number because 2 + 2 = 2 + 1 + 1.
The first few Smith numbers are:
Smith numbers were named by Albert Wilansky of Lehigh University
. He noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:
The number of Smith numbers below 10n for n=1,2,… is:
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 for n=1,2,… are:
Smith numbers can be constructed from factored repunit
s. The largest known Smith number is:
where R1031 is a repunit
equal to (101031−1)/9.
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
for which, in a given base
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
(in base 10
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
by default), the sum of its digits
Digit sum
In mathematics, the digit sum of a given integer is the sum of all its digits,...
is equal to the sum of the digits in its prime factor
Prime factor
In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...
ization. For example, 378 = 2 × 3 × 3 × 3 × 7 is a Smith number since 3 + 7 + 8 = 2 + 3 + 3 + 3 + 7. It's important to remember that, by definition, the factors are treated as digits. For example, 22 factors to 2 × 11 and yields three digits: 2, 1, 1. Therefore 22 is a Smith number because 2 + 2 = 2 + 1 + 1.
The first few Smith numbers 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, 1086, …
Smith numbers were named by Albert Wilansky of Lehigh University
Lehigh University
Lehigh University is a private, co-educational university located in Bethlehem, Pennsylvania, in the Lehigh Valley region of the United States. It was established in 1865 by Asa Packer as a four-year technical school, but has grown to include studies in a wide variety of disciplines...
. 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 = 42.
Properties
W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers.The number of Smith numbers below 10n 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. It is not known how many Smith brothers there are. The starting elements of the smallest Smith n-tuple for n=1,2,… are:
- 4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, …
Smith numbers can be constructed from factored repunit
Repunit
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler...
s. The largest known Smith number is:
- 9 × R1031 × (104594 + 3 + 1)1476
where R1031 is a repunit
Repunit
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler...
equal to (101031−1)/9.
External links
- Shyam Sunder Gupta, Fascinating Smith numbers.