Leyland number

In number theory , a Leyland number is a positive integer of the form ${\displaystyle n\in \mathbb {N} }$

${\displaystyle n=x^{y}+y^{x}}$with and and , ${\displaystyle x\in \mathbb {N} }$ ${\displaystyle y\in \mathbb {N} }$ ${\displaystyle x>1}$ ${\displaystyle y>1}$

If one were to forego the condition and , one could represent every natural number in the form whereby every number would be a Leyland number. ${\displaystyle x>1}$ ${\displaystyle y>1}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n=(n-1)^{1}+1^{n-1}}$

Sometimes one demands the additional condition so that one gets a clear representation of the Leyland numbers (otherwise one would have two slightly different representations with). ${\displaystyle x\geq y>1}$ ${\displaystyle 17=3^{2}+2^{3}=2^{3}+3^{2}}$

A prime Leyland number is called a Leyland prime number .

The Leyland numbers were named after the mathematician Paul Leyland ( en ).

Examples

• The first Leyland numbers are as follows:

8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124, 1649, 2169, 2530, 4240, 5392, 6250, 7073, 8361, 16580, 18785, 20412, 23401, 32993, 60049, 65792, 69632, 93312, 94932, 131361, 178478, 262468, 268705, 397585, 423393, 524649, 533169,… (follow A076980 in OEIS )

In the OEIS sequence A076980 above, the number that is displayed is also specified . Only because of this number is not a Leyland number. ${\displaystyle 3}$ ${\displaystyle 3=2^{1}+1^{2}}$ ${\displaystyle y=1\not >1}$

The first Leyland numbers have the following representation:

${\displaystyle 8=2^{2}+2^{2}}$, , , , , , , , ... ${\displaystyle 17=3^{2}+2^{3}}$ ${\displaystyle 32=4^{2}+2^{4}}$ ${\displaystyle 54=3^{3}+3^{3}}$ ${\displaystyle 57=5^{2}+2^{5}}$ ${\displaystyle 100=6^{2}+2^{6}}$ ${\displaystyle 145=4^{3}+3^{4}}$ ${\displaystyle 177=7^{2}+2^{7}}$

• The first Leyland primes are the following (again, the prime number is not one of them): ${\displaystyle 3}$

17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193, 4318114567396436564035293097707729426477458833, 5052785737795758503064406447721934417290878968063369478337, ... (sequence A094133 in OEIS )

The first Leyland prime numbers have the following representation:

${\displaystyle 17=3^{2}+2^{3}}$, , , , , , , , ... ${\displaystyle 593=9^{2}+2^{9}}$ ${\displaystyle 32993=15^{2}+2^{15}}$ ${\displaystyle 2097593=21^{2}+2^{21}}$ ${\displaystyle 8589935681=33^{2}+2^{33}}$ ${\displaystyle 59604644783353249=24^{5}+5^{24}}$ ${\displaystyle 523347633027360537213687137=56^{3}+3^{56}}$ ${\displaystyle 43143988327398957279342419750374600193=32^{15}+15^{32}}$

• If you leave the second base fixed, you get a Leyland prime number for the first base if and only if one of the following numbers is: ${\displaystyle y=2}$ ${\displaystyle x}$ ${\displaystyle x}$

3, 9, 15, 21, 33, 2007, 2127, 3759, 29355, 34653, 57285, 99069, ... (follow A064539 in OEIS )

These prime numbers thus all have the form . Again, the prime number does not actually belong to it because it is not a Leyland prime number because of it . ${\displaystyle p=x^{2}+2^{x}}$ ${\displaystyle 3=1^{2}+2^{1}}$ ${\displaystyle x=1\not >1}$

• The largest Leyland prime number up to November 2012 was . It was recognized as a prime number on October 15, 2010 with the program fastECPP . As a possible prime number ( probable prime , PRP ) she was known for some time. She has jobs. When it was discovered, it was the largest prime number found with elliptic curves to date (hence the name of the program: Elliptic Curve Primality Proving - ECPP ). ${\displaystyle 6753^{5122}+5122^{6753}}$ ${\displaystyle 25050}$
• On December 11, 2012, the currently (as of June 15, 2018) largest known Leyland prime number was discovered, namely . She has jobs. It was discovered as a possible prime number ( PRP ) by Anatoly F. Selevich, and it was recognized as a prime number with the CIDE program (by J. Franke, T. Kleinjung, A. Decker, J. Ecknig and A. Großwendt). ${\displaystyle 8656^{2929}+2929^{8656}}$ ${\displaystyle 30008}$
• There are at least 594 larger possible prime numbers that could be Leyland prime numbers. The currently largest are with digits (discovered by Anatoly F. Selevich) and with digits (discovered by Serge Batalov). They have already been recognized as possible prime numbers (PRP), but we still have to prove that they are actually prime numbers. ${\displaystyle 314738^{9}+9^{314738}}$ ${\displaystyle 300337}$ ${\displaystyle 328574^{15}+15^{328574}}$ ${\displaystyle 386434}$

application

Leyland primes do not have a suitable form by means of which one can determine with simple (known) algorithms whether they are prime or not. As mentioned above, it is relatively easy to determine that they are possible prime numbers ( PRP ), but to definitively prove the primality is very difficult. Therefore Leyland prime numbers are ideal test cases for general proofs of primality. For example, there are the Lucas test and the Pépin test for testing Fermat numbers with the form , which can test precisely such numbers for their primality particularly quickly. In the case of Leyland primes, there are no such tests specifically tailored to them. ${\displaystyle 2^{2^{n}}+1}$

Leyland numbers of the 2nd kind

In number theory, a Leyland number of the 2nd kind is a positive whole number of the form ${\displaystyle n\in \mathbb {N} }$

${\displaystyle n=x^{y}-y^{x}}$with and and , ${\displaystyle x\in \mathbb {N} }$ ${\displaystyle y\in \mathbb {N} }$ ${\displaystyle x>1}$ ${\displaystyle y>1}$

A prime Leyland number of the 2nd kind is called a Leyland prime number of the 2nd kind .

Examples

• The first Leyland numbers of the 2nd kind are the following:

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, 357857, 523927, 529713, 1038576, 1048176, ... ( continuation A045575 in OEIS )

The first Leyland numbers of the second kind have the following representation:

${\displaystyle 0=2^{2}-2^{2}}$, , , , , , , , , ... ${\displaystyle 1=3^{2}-2^{3}}$ ${\displaystyle 7=2^{5}-5^{2}}$ ${\displaystyle 17=3^{4}-4^{3}}$ ${\displaystyle 28=2^{6}-6^{2}}$ ${\displaystyle 79=2^{7}-7^{2}}$ ${\displaystyle 118=3^{5}-5^{3}}$ ${\displaystyle 192=2^{8}-8^{2}}$ ${\displaystyle 399=4^{5}-5^{4}}$

• The first Leyland primes of the 2nd kind are the following:

7, 17, 79, 431, 58049, 130783, 162287, 523927, 2486784401, 6102977801, 8375575711, 13055867207, 83695120256591, 375,700,268,413,577, 2251799813682647, 9007199254738183, 79792265017612001, 1490116119372884249 ... (sequence A123206 in OEIS )

The first Leyland primes of the 2nd kind have the following representation:

${\displaystyle 7=2^{5}-5^{2}}$, , , , , , , , ... ${\displaystyle 17=3^{4}-4^{3}}$ ${\displaystyle 79=2^{7}-7^{2}}$ ${\displaystyle 431=2^{9}-9^{2}}$ ${\displaystyle 58049=3^{10}-10^{3}}$ ${\displaystyle 130783=2^{17}-17^{2}}$ ${\displaystyle 162287=6^{7}-7^{6}}$ ${\displaystyle 523927=2^{19}-19^{2}}$

• The smallest Leyland prime numbers of the 2nd kind, i.e. of the form with increasing, are the following (where the value is if there is no such prime number): ${\displaystyle x^{y}-y^{x}}$ ${\displaystyle x=1,2,\ldots }$ ${\displaystyle 1}$

1 , 7, 17, 1 , 6102977801, 162287, 79792265017612001, 8375575711, 2486784401, ... ( continuation A122735 in OEIS )

The associated values ​​are as follows ${\displaystyle y}$

0, 5, 4, 0, 14, 7, 20, 11, 10, 273, 14, 13, 38, 89, 68, 0, ... (sequence A128355 in OEIS )

Examples: In the fifth position of the two upper series of numbers there is or , that is . In the fourth place is or , that is, there is no Leyland prime number of the form (because by definition it is not a Leyland prime number). ${\displaystyle 6102977801}$ ${\displaystyle 14}$ ${\displaystyle 6102977801=5^{14}-14^{5}}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 4^{y}-y^{4}}$ ${\displaystyle 4^{1}-1^{4}=3}$

Unsolved problem: From the 17th position of the values ​​of the OEIS sequence A128355, certain values ​​are not yet known. The values ​​are still unknown in the following places : ${\displaystyle y}$ ${\displaystyle y}$ ${\displaystyle y}$

17, 18, 22, 25, 26, 27, 28, ...

Example: It is still unknown whether there are prime numbers of the form or the form etc. ${\displaystyle x^{17}-17^{x}}$ ${\displaystyle x^{26}-26^{x}}$

• There are at least 1255 possible prime numbers, which could be Leyland prime numbers of the 2nd kind, which currently have PRP status. The currently largest are with posts (discovered by Serge Batalov) and with posts (discovered by Henri Lifchitz). They have already been recognized as possible prime numbers (PRP), but we still have to prove that they are actually prime numbers. ${\displaystyle 14^{161089}-161089^{14}}$ ${\displaystyle 184629}$ ${\displaystyle 2^{608541}+608541^{2}}$ ${\displaystyle 183190}$

properties

• Let the number of Leyland's numbers of the 2nd kind be smaller or equal . Then: ${\displaystyle N(x)}$ ${\displaystyle x}$

${\displaystyle N(x)\approx {\frac {(\log x)^{2}}{2\cdot (\log \log x)^{2}}}}$

Others

There is a project called "XYYXF" that deals with the factorization of possibly composite Leyland numbers. The same project deals with the factorization of possibly composite Leyland numbers of the 2nd kind.

Individual evidence

1. ^ Paul Leyland: Primes and Strong Pseudoprimes of the form x ^y + y ^x . (No longer available online.) Archived from the original on February 10, 2007 ; accessed on June 14, 2018 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
2. ↑ Chris K. Caldwell: The Largest Known Primes! 6753 ^ 5122 + 5122 ^ 6753. Prime Pages, accessed June 14, 2018 . 
3. ↑ Chris K. Caldwell: The Top Twenty: Elliptic Curve Primality Proof. Prime Pages, accessed June 14, 2018 . 
4. ↑ Mihailescu's CIDE. mersenneforum.org, accessed June 14, 2018 . 
5. ↑ ^{a b} Andrey Kulsha: Factorizations of x ^y + y ^x for 1 <y <x <151. mersenneforum.org, accessed June 14, 2018 . 
6. ^ ^{A b} Henri Lifchitz, Renaud Lifchitz: PRP Top Records - Search for: x ^y + y ^x . PRP Records, accessed June 14, 2018 . 
7. ^ ^{A b} Henri Lifchitz, Renaud Lifchitz: PRP Top Records - Search for: x ^y - y ^x . PRP Records, accessed June 15, 2018 . 
8. ↑ Neil JA Sloane : Comments on "Non-negative numbers of the form x ^ y - y ^ x, for x, y> 1." OEIS , accessed June 15, 2018 . 
9. ^ Michael Waldschmidt: Perfect Powers: Pillai's works and their developments. OEIS , accessed June 15, 2018 . 

Web links

• Leyland number . In: PlanetMath . (English)
• Leyland Numbers - Numberphile on YouTube