Euler Lucky Number/Examples/5
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Example of Euler Lucky Number
The expression:
- $n^2 + n + 5$
yields primes for $n = 0$ to $n = 3$.
This demonstrates that $5$ is a Euler lucky number.
Proof
\(\ds 0^2 + 0 + 5\) | \(=\) | \(\ds 0 + 0 + 5\) | \(\ds = 5\) | which is prime | ||||||||||
\(\ds 1^2 + 1 + 5\) | \(=\) | \(\ds 1 + 1 + 5\) | \(\ds = 7\) | which is prime | ||||||||||
\(\ds 2^2 + 2 + 5\) | \(=\) | \(\ds 4 + 2 + 5\) | \(\ds = 11\) | which is prime | ||||||||||
\(\ds 3^2 + 3 + 5\) | \(=\) | \(\ds 9 + 3 + 5\) | \(\ds = 17\) | which is prime |
This sequence is A027690 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
$\blacksquare$