mathematics

Leyland number

Leyland number is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Leyland number rather than just read about it. In short: In number theory, a Leyland number is a number of the form x y + y x {\displaystyle x^{y}+y^{x}} where x and y are integers greater than 1. They are named after the mathematician Paul Leyland.

Key takeaways

Leyland number belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Leyland number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Leyland number from memory before moving on to harder problems.

Reference excerpt

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

x y + y x {\displaystyle 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 (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 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. 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. 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. In February 2023, 1048245 + 5104824 (73269 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 3147389 + 9314738, 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

x y − y x {\displaystyle 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, ... (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).

References

External links Leyland Numbers - Numberphile on YouTube

Worked examples

Example 1 — a first encounter with Leyland number

Start with the simplest possible case. Write down what Leyland number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Leyland number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Leyland number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Leyland number

In research: Leyland number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Leyland number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Leyland number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Leyland number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Leyland number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Leyland number means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Leyland number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Leyland number in simple terms?

In number theory, a Leyland number is a number of the form x y + y x {\displaystyle x^{y}+y^{x}} where x and y are integers greater than 1. They are named after the mathematician Paul Leyland.

Why does Leyland number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Leyland number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Leyland number.