In
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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, a lucky number is a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
in a set which is generated by a certain "
sieve
A sieve, fine mesh strainer, or sift, is a device for separating wanted elements from unwanted material or for controlling the particle size distribution of a sample, using a screen such as a woven mesh or net or perforated sheet materia ...
". This sieve is similar to the
Sieve of Eratosthenes that generates the
primes
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 ...
, but it eliminates numbers based on their position in the remaining set, instead of their value (or position in the initial set of natural numbers).
The term was introduced in 1956 in a paper by Gardiner, Lazarus,
Metropolis
A metropolis () is a large city or conurbation which is a significant economic, political, and cultural center for a country or region, and an important hub for regional or international connections, commerce, and communications.
A big ci ...
and
Ulam. They suggest also calling its defining sieve, "the sieve of
Josephus
Flavius Josephus (; grc-gre, Ἰώσηπος, ; 37 – 100) was a first-century Romano-Jewish historian and military leader, best known for '' The Jewish War'', who was born in Jerusalem—then part of Roman Judea—to a father of priestly ...
Flavius" because of its similarity with the counting-out game in the
Josephus problem.
Lucky numbers share some properties with primes, such as asymptotic behaviour according to the
prime number theorem; also, a version of
Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. Twin lucky numbers and
twin prime
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...
s also appear to occur with similar frequency. However, if ''L''
''n'' denotes the ''n''-th lucky number, and ''p''
''n'' the ''n''-th prime, then ''L''
''n'' > ''p''
''n'' for all sufficiently large ''n''.
Because of their apparent similarites with the prime numbers, some mathematicians have suggested that some of their common properties may also be found in other sets of numbers generated by sieves of a certain unknown form, but there is little theoretical basis for this
conjecture.
The sieving process
Continue removing the ''n''th remaining numbers, where ''n'' is the next number in the list after the last surviving number. Next in this example is 9.
One way that the application of the procedure differs from that of the Sieve of Eratosthenes is that for ''n'' being the number being multiplied on a specific pass, the first number eliminated on the pass is the ''n''-th remaining number that has not yet been eliminated, as opposed to the number ''2n''. That is to say, the list of numbers this sieve counts through is different on each pass (for example 1, 3, 7, 9, 13, 15, 19... on the third pass), whereas in the Sieve of Eratosthenes, the sieve always counts through the entire original list (1, 2, 3...).
When this procedure has been carried out completely, the remaining integers are the lucky numbers (those that happen to be prime are in bold):
:
1,
3,
7,
9,
13,
15,
21,
25,
31,
33,
37,
43,
49,
51,
63,
67,
69,
73,
75,
79,
87,
93,
99,
105,
111,
115 115 may refer to:
* 115 (number), the number
* AD 115, a year in the 2nd century AD
* 115 BC, a year in the 2nd century BC
* 115 (Hampshire Fortress) Corps Engineer Regiment, Royal Engineers, a unit in the UK Territorial Army
* 115 (Leicestershire) ...
,
127,
129,
133,
135,
141 141 may refer to:
* 141 (number), an integer
* AD 141, a year of the Julian calendar
* 141 BC
__NOTOC__
Year 141 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Caepio and Pompeius (or ...
,
151,
159,
163,
169,
171,
189,
193,
195
Year 195 ( CXCV) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Scrapula and Clemens (or, less frequently, year 948 ''Ab urbe cond ...
,
201
Year 201 ( CCI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Fabianus and Arrius (or, less frequently, year 954 ''Ab urbe condit ...
,
205,
211
Year 211 ( CCXI) was a common year starting on Tuesday of the Julian calendar. At the time, in the Roman Empire it was known as the Year of the Consulship of Terentius and Bassus (or, less frequently, year 964 ''Ab urbe condita''). The denomin ...
,
219,
223
__NOTOC__
Year 223 (Roman numerals, CCXXIII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Maximus and Aelianus (or, less frequen ...
,
231,
235,
237,
241
Year 241 ( CCXLI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Gordianus and Pompeianus by the Romans (or, less frequently, year 9 ...
,
259
Year 259 ( CCLIX) 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 Aemilianus and Bassus (or, less frequently, year 1012 ''Ab urbe co ...
,
261,
267,
273,
283,
285,
289,
297,
303,
307,
319,
321,
327, ... .
The lucky number which removes ''n'' from the list of lucky numbers is: (0 if ''n'' is a lucky number)
:0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 7, 2, 0, 2, 3, 2, 0, 2, 9, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 7, 2, 3, 2, 0, 2, 13, 2, 3, 2, 0, 2, 0, 2, 3, 2, 15, 2, 9, 2, 3, 2, 7, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 7, 2, 3, 2, 21, 2, ...
Lucky primes
A "lucky prime" is a lucky number that is prime. They are:
:3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997, ... .
It has been conjectured that there are infinitely many lucky primes.
See also
*
Lucky numbers of Euler Euler's "lucky" numbers are positive integers ''n'' such that for all integers ''k'' with , the polynomial produces a prime number.
When ''k'' is equal to ''n'', the value cannot be prime since is divisible by ''n''. Since the polynomial can b ...
*
Fortunate number
A Fortunate number, named after Reo Fortune, is the smallest integer ''m'' > 1 such that, for a given positive integer ''n'', ''p'n''# + ''m'' is a prime number, where the primorial ''p'n''# is the product of the first ''n'' prime numbers.
...
*
Happy number
In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy number because ...
*
Harshad number
*
Gambling
Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three el ...
*
Lottery
*
Keno
References
Further reading
*
External links
Lucky Numbersby Enrique Zeleny,
The Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
*
{{Authority control
Integer sequences