mathematics

Lucky numbers of Euler

Lucky numbers of Euler 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 Lucky numbers of Euler rather than just read about it. In short: Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. Characteristics When k is equal to n, the value cannot be prime since n2 − n + n = n2 is divisible by n.

Key takeaways

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

Reference excerpt

Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number.

Characteristics When k is equal to n, the value cannot be prime since n2 − n + n = n2 is divisible by n. Since the polynomial can be written as k(k−1) + n, using the integers k with −(n−1) < k ≤ 0 produces the same set of numbers as 1 ≤ k < n. These polynomials are all members of the larger set of prime generating polynomials. Leonhard Euler published the polynomial k2 − k + 41 which produces prime numbers for all integer values of k from 1 to 40. As established by Kurt Heegner in 1952, only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS). Note that these numbers are all prime numbers. The primes of the form k2 − k + 41 are

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, ... (sequence A005846 in the OEIS).

Other lucky numbers Euler's lucky numbers are unrelated to the "lucky numbers" defined by a sieve algorithm. In fact, the only number which is both lucky and Euler-lucky is 3, since all other Euler-lucky numbers are congruent to 2 modulo 3, but no lucky numbers are congruent to 2 modulo 3.

See also Heegner number List of topics named after Leonhard Euler Formula for primes Ulam spiral

References

Literature Le Lionnais, F. Les Nombres Remarquables. Paris: Hermann, pp. 88 and 144, 1983. Leonhard Euler, Extrait d'un lettre de M. Euler le pere à M. Bernoulli concernant le Mémoire imprimé parmi ceux de 1771, p. 318 (1774). Euler Archive - All Works. 461.

External links Weisstein, Eric W. "Lucky Number of Euler". MathWorld.

Worked examples

Example 1 — a first encounter with Lucky numbers of Euler

Start with the simplest possible case. Write down what Lucky numbers of Euler 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 Lucky numbers of Euler 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 Lucky numbers of Euler 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 Lucky numbers of Euler

In research: Lucky numbers of Euler 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 Lucky numbers of Euler 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: Lucky numbers of Euler is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, Leonhard Euler, Prime numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Lucky numbers of Euler 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 Lucky numbers of Euler in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lucky numbers of Euler 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 Lucky numbers of Euler out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lucky numbers of Euler in simple terms?

Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. Characteristics When k is equal to n, the value cannot be prime since n2 − n + n = n2 is divisible by n.

Why does Lucky numbers of Euler 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 Lucky numbers of Euler?

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 Lucky numbers of Euler.