Arithmetic Progression of 16 Primes

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Theorem

The $16$ integers in arithmetic progression 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 progression of primes does not extend to $n < 0$.


\(\displaystyle 2\,236\,133\,941 + 0 \times 223\,092\,870\) \(=\) \(\displaystyle 2\,236\,133\,941\) which is prime
\(\displaystyle 2\,236\,133\,941 + 1 \times 223\,092\,870\) \(=\) \(\displaystyle 2\,459\,226\,811\) which is prime
\(\displaystyle 2\,236\,133\,941 + 2 \times 223\,092\,870\) \(=\) \(\displaystyle 2\,682\,319\,681\) which is prime
\(\displaystyle 2\,236\,133\,941 + 3 \times 223\,092\,870\) \(=\) \(\displaystyle 2\,905\,412\,551\) which is prime
\(\displaystyle 2\,236\,133\,941 + 4 \times 223\,092\,870\) \(=\) \(\displaystyle 3\,128\,505\,421\) which is prime
\(\displaystyle 2\,236\,133\,941 + 5 \times 223\,092\,870\) \(=\) \(\displaystyle 3\,351\,598\,291\) which is prime
\(\displaystyle 2\,236\,133\,941 + 6 \times 223\,092\,870\) \(=\) \(\displaystyle 3\,574\,691\,161\) which is prime
\(\displaystyle 2\,236\,133\,941 + 7 \times 223\,092\,870\) \(=\) \(\displaystyle 3\,797\,784\,031\) which is prime
\(\displaystyle 2\,236\,133\,941 + 8 \times 223\,092\,870\) \(=\) \(\displaystyle 4\,020\,876\,901\) which is prime
\(\displaystyle 2\,236\,133\,941 + 9 \times 223\,092\,870\) \(=\) \(\displaystyle 4\,243\,969\,771\) which is prime
\(\displaystyle 2\,236\,133\,941 + 10 \times 223\,092\,870\) \(=\) \(\displaystyle 4\,467\,062\,641\) which is prime
\(\displaystyle 2\,236\,133\,941 + 11 \times 223\,092\,870\) \(=\) \(\displaystyle 4\,690\,155\,511\) which is prime
\(\displaystyle 2\,236\,133\,941 + 12 \times 223\,092\,870\) \(=\) \(\displaystyle 4\,913\,248\,381\) which is prime
\(\displaystyle 2\,236\,133\,941 + 13 \times 223\,092\,870\) \(=\) \(\displaystyle 5\,136\,341\,251\) which is prime
\(\displaystyle 2\,236\,133\,941 + 14 \times 223\,092\,870\) \(=\) \(\displaystyle 5\,359\,434\,121\) which is prime
\(\displaystyle 2\,236\,133\,941 + 15 \times 223\,092\,870\) \(=\) \(\displaystyle 5\,582\,526\,991\) 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