Leyland number
In number theory, a Leyland number is a number of the form
where x and y are integers greater than 1.[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 (sequence A076980 in the OEIS).
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).
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, ... (sequence A094133 in the OEIS)
corresponding to
- 32+23, 92+29, 152+215, 212+221, 332+233, 245+524, 563+356, 3215+1532.[2]
One can also fix the value of y and consider the sequence of x values that gives Leyland primes, for example x2 + 2x is prime for x = 3, 9, 15, 21, 33, 2007, 2127, 3759, ... (OEIS: A064539).
By November 2012, the largest Leyland number that had been proven to be prime was 51226753 + 67535122 with 25050 digits. From January 2011 to April 2011, it was the largest prime whose primality was proved by elliptic curve primality proving.[3] In December 2012, this was improved by proving the primality of the two numbers 311063 + 633110 (5596 digits) and 86562929 + 29298656 (30008 digits), the latter of which surpassed the previous record.[4] In February 2023, 1048245 + 5104824 (73269 digits) was proven to be prime,[5] and it was also the largest prime proven using ECPP, until three months later a larger (non-Leyland) prime was proven using ECPP.[6] There are many larger known probable primes such as 3147389 + 9314738,[7] 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.[8]
Leyland number of the second kind
A Leyland number of the second kind is a number of the form
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, ... (sequence A045575 in the OEIS)
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, ... (sequence A123206 in the OEIS)
For the probable primes, see Henri Lifchitz & Renaud Lifchitz, PRP Top Records search.[7]
References
- ^ Richard Crandall and Carl Pomerance (2005), Prime Numbers: A Computational Perspective, Springer
- ^ "Primes and Strong Pseudoprimes of the form xy + yx". Paul Leyland. Archived from the original on 2007-02-10. Retrieved 2007-01-14.
- ^ "Elliptic Curve Primality Proof". Chris Caldwell. Retrieved 2011-04-03.
- ^ "Mihailescu's CIDE". mersenneforum.org. 2012-12-11. Retrieved 2012-12-26.
- ^ "Leyland prime of the form 1048245+5104824". Prime Wiki. Retrieved 2023-11-26.
- ^ "Elliptic Curve Primality Proof". Prime Pages. Retrieved 2023-11-26.
- ^ a b Henri Lifchitz & Renaud Lifchitz, PRP Top Records search.
- ^ "Factorizations of xy + yx for 1 < y < x < 151". Andrey Kulsha. Retrieved 2008-06-24.
External links
- Leyland Numbers - Numberphile on YouTube
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- Fermat (22n + 1)
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- Wagstaff (2p + 1)/3
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