Euler's "lucky" numbers are
positive integers ''n'' such that for all integers ''k'' with , the polynomial produces a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
.
When ''k'' is equal to ''n'', the value cannot be prime since is
divisible by ''n''. Since the polynomial can be written as , using the integers ''k'' with produces the same
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of numbers as . These polynomials are all members of the larger set of prime generating polynomials.
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
published the polynomial which produces prime numbers for all integer values of ''k'' from 1 to 40. Only 7 lucky numbers of Euler exist, namely 1, 2, 3, 5, 11, 17 and 41 . Note that these numbers are all prime numbers except for 1.
The primes of the form ''k''
2 − ''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, ... .
[See also the sieve algorithm for all such primes: ]
Euler's lucky numbers are unrelated to the "
lucky number
In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the rema ...
s" 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
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
3, but no lucky numbers are congruent to 2 modulo 3.
See also
*
Heegner number In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factori ...
*
List of topics named after Leonhard Euler
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler inclu ...
*
Formula for primes
*
Ulam spiral
References
Literature
*
Le Lionnais, F. ''Les Nombres Remarquables''. Paris: Hermann, pp. 88 and 144, 1983.
*
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
''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
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Integer sequences
Prime numbers
Leonhard Euler
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