Arithmetic Sequence of 16 Primes
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Theorem
The $16$ integers in arithmetic sequence defined as:
- $2\,236\,133\,941 + 223\,092\,870 n$
are prime for $n = 0, 1, \ldots, 15$.
Proof
First we note that:
- $2\,236\,133\,941 - 223\,092\,870 = 2\,013\,041\,071 = 53 \times 89 \times 426\,763$
and so this arithmetic sequence of primes does not extend to $n < 0$.
\(\ds 2\,236\,133\,941 + 0 \times 223\,092\,870\) | \(=\) | \(\ds 2\,236\,133\,941\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 1 \times 223\,092\,870\) | \(=\) | \(\ds 2\,459\,226\,811\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 2 \times 223\,092\,870\) | \(=\) | \(\ds 2\,682\,319\,681\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 3 \times 223\,092\,870\) | \(=\) | \(\ds 2\,905\,412\,551\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 4 \times 223\,092\,870\) | \(=\) | \(\ds 3\,128\,505\,421\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 5 \times 223\,092\,870\) | \(=\) | \(\ds 3\,351\,598\,291\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 6 \times 223\,092\,870\) | \(=\) | \(\ds 3\,574\,691\,161\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 7 \times 223\,092\,870\) | \(=\) | \(\ds 3\,797\,784\,031\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 8 \times 223\,092\,870\) | \(=\) | \(\ds 4\,020\,876\,901\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 9 \times 223\,092\,870\) | \(=\) | \(\ds 4\,243\,969\,771\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 10 \times 223\,092\,870\) | \(=\) | \(\ds 4\,467\,062\,641\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 11 \times 223\,092\,870\) | \(=\) | \(\ds 4\,690\,155\,511\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 12 \times 223\,092\,870\) | \(=\) | \(\ds 4\,913\,248\,381\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 13 \times 223\,092\,870\) | \(=\) | \(\ds 5\,136\,341\,251\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 14 \times 223\,092\,870\) | \(=\) | \(\ds 5\,359\,434\,121\) | which is prime | |||||||||||
\(\ds 2\,236\,133\,941 + 15 \times 223\,092\,870\) | \(=\) | \(\ds 5\,582\,526\,991\) | which is prime |
But note that $2\,236\,133\,941 + 16 \times 223\,092\,870 = 5\,805\,619\,861 = 79 \times 73\,488\,859$ and so is not prime.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2,236,133,941$