Leyland number explained

In number theory, a Leyland number is a number of the form

x ^{y} + y ^{x} where x and y are integers 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 x¹ + 1^x. 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).

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

3²+2³, 9²+2⁹, 15²+2¹⁵, 21²+2²¹, 33²+2³³, 24⁵+5²⁴, 56³+3⁵⁶, 32¹⁵+15³².^[1]

One can also fix the value of y and consider the sequence of x values that gives Leyland primes, for example x² + 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 5122⁶⁷⁵³ + 6753⁵¹²² with digits. From January 2011 to April 2011, it was the largest prime whose primality was proved by elliptic curve primality proving.^[2] In December 2012, this was improved by proving the primality of the two numbers 3110⁶³ + 63³¹¹⁰ (5596 digits) and 8656²⁹²⁹ + 2929⁸⁶⁵⁶ (digits), the latter of which surpassed the previous record.^[3] In February 2023, 104824⁵ + 5¹⁰⁴⁸²⁴ (digits) was proven to be prime,^[4] and it was also the largest prime proven using ECPP, until three months later a larger (non-Leyland) prime was proven using ECPP.^[5] There are many larger known probable primes such as 314738⁹ + 9³¹⁴⁷³⁸,^[6] 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.^[7]

Leyland number of the second kind

A Leyland number of the second kind is a number of the form

x ^{y} - y ^{x} where x and y are integers 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.

Notes and References

1. Web site: Primes and Strong Pseudoprimes of the form x^y + y^x . Paul Leyland . 2007-01-14 . https://web.archive.org/web/20070210024511/http://www.leyland.vispa.com/numth/primes/xyyx.htm . 2007-02-10 . dead .
2. Web site: Elliptic Curve Primality Proof . Chris Caldwell . 2011-04-03.
3. Web site: Mihailescu's CIDE . mersenneforum.org . 2012-12-11 . 2012-12-26.
4. Web site: Leyland prime of the form 104824⁵+5¹⁰⁴⁸²⁴ . Prime Wiki . 2023-11-26.
5. Web site: Elliptic Curve Primality Proof . Prime Pages . 2023-11-26.
6. Henri Lifchitz & Renaud Lifchitz, PRP Top Records search.
7. Web site: Factorizations of x^y + y^x for 1 < y < x < 151 . Andrey Kulsha . 2008-06-24.