Euler's "lucky" numbers are positive integers n such that m² − m + n is a prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 for m = 0, …, n − 1.

Leonhard Euler

Leonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 published the polynomial x² − x + 41 which produces prime numbers for all integer values of x from 0 to 40. Obviously, when x is equal to 41, the value cannot be prime any more since it is divisible by 41. Only 6 numbers have this property, namely 2, 3, 5, 11, 17 and 41 .

These numbers are not related to the so-called lucky number

Lucky number

In number theory, a lucky number is a natural number in a set which is generated by a "sieve" similar to the Sieve of Eratosthenes that generates the primes.Begin with a list of integers starting with 1:...

s.

See also

• Heegner number
  Heegner number
  In number theory, a Heegner number is a square-free positive integer d such that the imaginary quadratic field Q has class number 1...
  
  
• List of topics named after Leonhard Euler
• Formula for primes
  Formula for primes
  In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is easily computable is presently known...