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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, a Leyland number is a number of the form
:
where ''x'' and ''y'' are
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s greater than 1.
They are named after the mathematician
Paul Leyland
Paul Leyland is a British number theorist who has studied integer factorization and primality testing.
He has contributed to the factorization of RSA-129, RSA-140, and RSA-155, as well as potential factorial primes as large as 400! + 1. He has ...
. The first few Leyland numbers are
:
8,
17,
32,
54,
57,
100,
145 145 may refer to:
*145 (number), a natural number
*AD 145, a year in the 2nd century AD
* 145 BC, a year in the 2nd century BC
*145 (dinghy), a two-person intermediate sailing dinghy
* 145 (South) Brigade
* 145 (New Jersey bus)
See also
* List of ...
,
177
Year 177 ( CLXXVII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Commodus and Plautius (or, less frequently, year 930 ''Ab urbe co ...
,
320,
368
Year 368 ( CCCLXVIII) was a leap year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Augustus and Valens (or, less frequently, year 1121 ''Ab urbe con ...
,
512,
593,
945
Year 945 ( CMXLV) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* January 27 – The co-emperors Stephen and Constantine are overthrown barely ...
,
1124 .
The requirement that ''x'' and ''y'' both be greater than 1 is important, since without it every positive integer would be a Leyland number of the form ''x''
1 + 1
''x''. Also, because of the
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
property of addition, the condition ''x'' ≥ ''y'' is usually added to avoid double-covering the set of Leyland numbers (so we have 1 < ''y'' ≤ ''x'').
Leyland primes
A Leyland prime is a Leyland number that is also a prime. The first such primes are:
:
17,
593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193, ...
corresponding to
:3
2+2
3, 9
2+2
9, 15
2+2
15, 21
2+2
21, 33
2+2
33, 24
5+5
24, 56
3+3
56, 32
15+15
32.
One can also fix the value of ''y'' and consider the sequence of ''x'' values that gives Leyland primes, for example ''x''
2 + 2
''x'' is prime for ''x'' = 3, 9, 15, 21, 33, 2007, 2127, 3759, ... ().
By November 2012, the largest Leyland number that had been proven to be prime was 5122
6753 + 6753
5122 with 25050 digits. From January 2011 to April 2011, it was the largest prime whose primality was proved by
elliptic curve primality proving In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 a ...
.
In December 2012, this was improved by proving the primality of the two numbers 3110
63 + 63
3110 (5596 digits) and 8656
2929 + 2929
8656 (30008 digits), the latter of which surpassed the previous record. There are many larger known
probable prime
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific con ...
s such as 314738
9 + 9
314738, but it is hard to prove primality of large Leyland numbers.
Paul Leyland
Paul Leyland is a British number theorist who has studied integer factorization and primality testing.
He has contributed to the factorization of RSA-129, RSA-140, and RSA-155, as well as potential factorial primes as large as 400! + 1. He has ...
writes on his website: "More recently still, it was realized that numbers of this form are ideal test cases for general purpose primality proving programs. They have a simple algebraic description but no obvious
cyclotomic
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because o ...
properties which special purpose algorithms can exploit."
There is a project called XYYXF to
factor
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, suc ...
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...
Leyland numbers.
Leyland number of the second kind
A Leyland number of the second kind is a number of the form
:
where ''x'' and ''y'' are
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s greater than 1. The first such numbers are:
: 0, 1,
7,
17, 28,
79, 118, 192, 399,
431
Year 431 (Roman numerals, CDXXXI) 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 Bassus and Antiochus (or, less frequently, year 1 ...
, 513, 924, 1844, 1927, 2800, 3952, 6049, 7849, 8023, 13983, 16188, 18954, 32543, 58049, 61318, 61440, 65280, 130783, 162287, 175816, 255583, 261820, ...
A Leyland prime of the second kind is a Leyland number of the second kind that is also prime. The first few such primes are:
:7, 17, 79, 431, 58049, 130783, 162287, 523927, 2486784401, 6102977801, 8375575711, 13055867207, 83695120256591, 375700268413577, 2251799813682647, ...
For the probable primes, see Henri Lifchitz & Renaud Lifchitz, PRP Top Records search.
[Henri Lifchitz & Renaud Lifchitz]
PRP Top Records search
/ref>
References
External links
*
{{DEFAULTSORT:Leyland Number
Integer sequences