Prime-Generating Quadratic of form x squared - 79 x + 1601

Theorem

The quadratic function:

$x^2 - 79 x + 1601$

gives prime values for integer $x$ such that $0 \le x \le 79$.

The primes generated are repeated once each.

Proof

Let $x = z + 40$.

Then:

\(\ds \)

\(\)

\(\ds \left({z + 40}\right)^2 - 79 \left({z + 40}\right) + 1601\)

\(\ds \)

\(=\)

\(\ds z^2 + 2 \times 40 z + 40^2 - 79 z - 79 \times 40 + 1601\)

\(\ds \)

\(=\)

\(\ds z^2 + 80 z + 1600 - 79 z - 3160 + 1601\)

\(\ds \)

\(=\)

\(\ds z^2 + z + 41\)

Thus it can be seen that this is an application of Euler Lucky Number $41$.

$\blacksquare$

Sources

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