# Euler Lucky Number/Examples/41

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## Example of Euler Lucky Number

The expression:

- $n^2 + n + 41$

yields primes for $n = 0$ to $n = 39$.

It also generates the same set of primes for $n = -1 \to n = -40$.

These are not the only primes generated by this formula.

No other quadratic function of the form $x^2 + a x + b$, where $a, b \in \Z_{>0}$ and $a, b < 10000$ generates a longer sequence of primes.

## Proof

\(\ds 0^2 + 0 + 41\) | \(=\) | \(\ds 0 + 0 + 41\) | \(\ds = 41\) | which is prime | ||||||||||

\(\ds 1^2 + 1 + 41\) | \(=\) | \(\ds 1 + 1 + 41\) | \(\ds = 43\) | which is prime | ||||||||||

\(\ds 2^2 + 2 + 41\) | \(=\) | \(\ds 4 + 2 + 41\) | \(\ds = 47\) | which is prime | ||||||||||

\(\ds 3^2 + 3 + 41\) | \(=\) | \(\ds 9 + 3 + 41\) | \(\ds = 53\) | which is prime | ||||||||||

\(\ds 4^2 + 4 + 41\) | \(=\) | \(\ds 16 + 4 + 41\) | \(\ds = 61\) | which is prime | ||||||||||

\(\ds 5^2 + 5 + 41\) | \(=\) | \(\ds 25 + 5 + 41\) | \(\ds = 71\) | which is prime | ||||||||||

\(\ds 6^2 + 6 + 41\) | \(=\) | \(\ds 36 + 6 + 41\) | \(\ds = 83\) | which is prime | ||||||||||

\(\ds 7^2 + 7 + 41\) | \(=\) | \(\ds 49 + 7 + 41\) | \(\ds = 97\) | which is prime | ||||||||||

\(\ds 8^2 + 8 + 41\) | \(=\) | \(\ds 64 + 8 + 41\) | \(\ds = 113\) | which is prime | ||||||||||

\(\ds 9^2 + 9 + 41\) | \(=\) | \(\ds 81 + 9 + 41\) | \(\ds = 131\) | which is prime | ||||||||||

\(\ds 10^2 + 10 + 41\) | \(=\) | \(\ds 100 + 10 + 41\) | \(\ds = 151\) | which is prime | ||||||||||

\(\ds 11^2 + 11 + 17\) | \(=\) | \(\ds 121 + 11 + 17\) | \(\ds = 173\) | which is prime | ||||||||||

\(\ds 12^2 + 12 + 41\) | \(=\) | \(\ds 144 + 12 + 41\) | \(\ds = 197\) | which is prime | ||||||||||

\(\ds 13^2 + 13 + 17\) | \(=\) | \(\ds 169 + 13 + 17\) | \(\ds = 223\) | which is prime | ||||||||||

\(\ds 14^2 + 14 + 41\) | \(=\) | \(\ds 196 + 14 + 41\) | \(\ds = 251\) | which is prime | ||||||||||

\(\ds 15^2 + 15 + 41\) | \(=\) | \(\ds 225 + 15 + 41\) | \(\ds = 281\) | which is prime | ||||||||||

\(\ds 16^2 + 16 + 41\) | \(=\) | \(\ds 256 + 16 + 41\) | \(\ds = 313\) | which is prime | ||||||||||

\(\ds 17^2 + 17 + 41\) | \(=\) | \(\ds 289 + 17 + 41\) | \(\ds = 347\) | which is prime | ||||||||||

\(\ds 18^2 + 18 + 41\) | \(=\) | \(\ds 324 + 18 + 41\) | \(\ds = 383\) | which is prime | ||||||||||

\(\ds 19^2 + 19 + 41\) | \(=\) | \(\ds 361 + 19 + 41\) | \(\ds = 421\) | which is prime | ||||||||||

\(\ds 20^2 + 20 + 41\) | \(=\) | \(\ds 400 + 20 + 41\) | \(\ds = 461\) | which is prime | ||||||||||

\(\ds 21^2 + 21 + 41\) | \(=\) | \(\ds 441 + 21 + 41\) | \(\ds = 503\) | which is prime | ||||||||||

\(\ds 22^2 + 22 + 41\) | \(=\) | \(\ds 484 + 22 + 41\) | \(\ds = 547\) | which is prime | ||||||||||

\(\ds 23^2 + 23 + 41\) | \(=\) | \(\ds 529 + 23 + 41\) | \(\ds = 593\) | which is prime | ||||||||||

\(\ds 24^2 + 24 + 41\) | \(=\) | \(\ds 576 + 24 + 41\) | \(\ds = 641\) | which is prime | ||||||||||

\(\ds 25^2 + 25 + 41\) | \(=\) | \(\ds 625 + 25 + 41\) | \(\ds = 691\) | which is prime | ||||||||||

\(\ds 26^2 + 26 + 41\) | \(=\) | \(\ds 676 + 26 + 41\) | \(\ds = 743\) | which is prime | ||||||||||

\(\ds 27^2 + 27 + 41\) | \(=\) | \(\ds 729 + 27 + 41\) | \(\ds = 797\) | which is prime | ||||||||||

\(\ds 28^2 + 28 + 41\) | \(=\) | \(\ds 784 + 28 + 41\) | \(\ds = 853\) | which is prime | ||||||||||

\(\ds 29^2 + 29 + 41\) | \(=\) | \(\ds 841 + 29 + 41\) | \(\ds = 911\) | which is prime | ||||||||||

\(\ds 30^2 + 30 + 41\) | \(=\) | \(\ds 900 + 30 + 41\) | \(\ds = 971\) | which is prime | ||||||||||

\(\ds 31^2 + 31 + 41\) | \(=\) | \(\ds 961 + 31 + 41\) | \(\ds = 1033\) | which is prime | ||||||||||

\(\ds 32^2 + 32 + 41\) | \(=\) | \(\ds 1024 + 32 + 41\) | \(\ds = 1097\) | which is prime | ||||||||||

\(\ds 33^2 + 33 + 41\) | \(=\) | \(\ds 1089 + 33 + 41\) | \(\ds = 1163\) | which is prime | ||||||||||

\(\ds 34^2 + 34 + 41\) | \(=\) | \(\ds 1156 + 34 + 41\) | \(\ds = 1231\) | which is prime | ||||||||||

\(\ds 35^2 + 35 + 41\) | \(=\) | \(\ds 1225 + 35 + 41\) | \(\ds = 1301\) | which is prime | ||||||||||

\(\ds 36^2 + 36 + 41\) | \(=\) | \(\ds 1296 + 36 + 41\) | \(\ds = 1373\) | which is prime | ||||||||||

\(\ds 37^2 + 37 + 41\) | \(=\) | \(\ds 1369 + 37 + 41\) | \(\ds = 1447\) | which is prime | ||||||||||

\(\ds 38^2 + 38 + 41\) | \(=\) | \(\ds 1444 + 38 + 41\) | \(\ds = 1523\) | which is prime | ||||||||||

\(\ds 39^2 + 39 + 41\) | \(=\) | \(\ds 1521 + 39 + 41\) | \(\ds = 1601\) | which is prime | ||||||||||

\(\ds 40^2 + 40 + 41\) | \(=\) | \(\ds 1600 + 40+ 41\) | \(\ds = 1681\) | which is not prime: $1681 = 41^2$ |

This sequence is A005846 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Then we have:

\(\ds \paren {-\paren {n + 1} }^2 + \paren {-\paren {n + 1} }\) | \(=\) | \(\ds n^2 + 2 n + 1 - \paren {n + 1}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds n^2 + n\) |

and so replacing $0$ to $39$ with $-1$ to $-40$ yields exactly the same sequence of primes.

We note in addition the example:

- $581^2 + 581 + 41 = 338 \, 183$

which is prime.

$\blacksquare$

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $41$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $41$