In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, a Leyland number is a number of the form
:
where ''x'' and ''y'' are
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s greater than 1.
They are named after the mathematician
Paul Leyland. The first few Leyland numbers are
:
8,
17,
32,
54,
57,
100,
145,
177,
320,
368,
512,
593,
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 ''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. Perhaps most familiar as a pr ...
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
prime
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 ...
. 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 digits. From January 2011 to April 2011, it was the largest prime whose primality was proved by
elliptic curve primality proving.
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 ( digits), the latter of which surpassed the previous record. In February 2023, 104824
5 + 5
104824 ( digits) was proven to be prime, and it was also the largest prime proven using ECPP, until three months later a larger (non-Leyland) prime was proven using ECPP.
There are many larger known
probable primes such as 314738
9 + 9
314738,
[Henri Lifchitz & Renaud Lifchitz]
PRP Top Records search
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.
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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s greater than 1. The first such numbers are:
: 0, 1,
7,
17,
28,
79,
118,
192,
399,
431, 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, ... . We can also consider 145 in the form of 4 to the power of 3 plus 4 to the power of 4.
For the probable primes, see Henri Lifchitz & Renaud Lifchitz, PRP Top Records search.
References
External links
*
{{DEFAULTSORT:Leyland Number
Eponymous numbers in mathematics
Integer sequences