Leyland number
Encyclopedia
In number theory
, a Leyland number is a number of the form xy + yx, where x and y are integer
s greater than 1. The first few Leyland numbers are
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 x1 + 1x. Also, because of the commutative
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).
The first prime
Leyland numbers are
corresponding to
As of June 2008, the largest Leyland number that has been proven to be prime is 26384405 + 44052638 with 15071 digits. From July 2004 to June 2006, it was the largest prime whose primality was proved by elliptic curve primality proving
. There are many larger known probable prime
s such as 913829 + 991382, but it is hard to prove primality of large Leyland numbers. Paul Leyland
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 properties which special purpose algorithms can exploit."
There is a project called XYYXF to factor
composite
Leyland numbers.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a Leyland number is a number of the form xy + yx, where x and y are integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s greater than 1. The first few Leyland numbers are
- 8, 1717 (number)17 is the natural number following 16 and preceding 18. It is prime.In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed: 17 vs 70...
, 3232 (number)32 is the natural number following 31 and preceding 33.-In mathematics:32 is the smallest number n with exactly 7 solutions to the equation φ = n...
, 5454 (number)54 is the natural number following 53 and preceding 55.-In mathematics:54 is a 19-gonal number. Twice the third power of three, 54 is a Leyland number. 54 can be written as the sum of three squares in three different ways: 7^2 + 2^2 + 1^2 = 6^2 + 2 = 2 + 2^2 = 54. It is the smallest number with...
, 5757 (number)57 is the natural number following 56 and preceding 58.- In mathematics :Fifty-seven is the sixteenth discrete semiprime and the sixth in the family. With 58 it forms the fourth discrete bi-prime pair. 57 has an aliquot sum of 23 and is the first composite member of the 23-aliquot tree...
, 100100 (number)100 is the natural number following 99 and preceding 101.-In mathematics:One hundred is the square of 10...
, 145145 (number)145 is the natural number following 144 and preceding 146.- In mathematics :* Although composite, 145 is a pseudoprime.* Given 145, the Mertens function returns 0.* 145 is a pentagonal number and a centered square number....
, 177177 (number)177 is the natural number following 176 and preceding 178.-In mathematics:* 177 is an odd number* 177 is a composite number* 177 is a deficient number, as 63 is less than 177* 177 is a Leyland number since it can be expressed as 27 + 72...
, 320, 368, 512512 (number)512 is the natural number following 511 and preceding 513.512 is a power of two: 29 and the cube of 8: 83.Also, it is the eleventh Leyland number.- Special use in computers :...
, 593593 (number)593 is the natural number following 592 and preceding 594.-In mathematics:593 is an odd number. It is a prime number, an example of what Paul Erdős and Ernst G. Straus called a Good prime, or a prime whose square is greater than the product of its neighboring two primes. As such it is part of...
, 945, 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 x1 + 1x. Also, because of the commutative
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
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).
The first prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
Leyland numbers are
- 17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193
corresponding to
- 32+23, 92+29, 152+215, 212+221, 332+233, 245+524, 563+356, 3215+1532.
As of June 2008, the largest Leyland number that has been proven to be prime is 26384405 + 44052638 with 15071 digits. From July 2004 to June 2006, it was the largest prime whose primality was proved by elliptic curve primality proving
Elliptic curve primality proving
Elliptic Curve Primality Proving is a method based on elliptic curves to prove the primality of a number . It is a general-purpose algorithm, meaning it does not depend on the number being of a special form...
. There are many larger known probable prime
Probable prime
In number theory, a probable prime is an integer that satisfies a specific condition also satisfied by all prime numbers. Different types of probable primes have different specific conditions...
s such as 913829 + 991382, but it is hard to prove primality of large Leyland numbers. Paul Leyland
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 also studied Cunningham numbers, Cullen numbers, Woodall...
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 properties which special purpose algorithms can exploit."
There is a project called XYYXF to factor
Integer factorization
In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....
composite
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....
Leyland numbers.