Sequence of 9 Primes of form 4n+1

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Theorem

The following sequence of $9$ consecutive prime numbers are all of the form $4 n + 1$:

$11 \, 593, 11 \, 597, 11 \, 617, 11 \, 621, 11 \, 633, 11 \, 657, 11 \, 677, 11 \, 681, 11 \, 689$


Proof

\(\ds 11 \, 593\) \(=\) \(\ds 4 \times 2898 + 1\)
\(\ds 11 \, 597\) \(=\) \(\ds 4 \times 2899 + 1\)
\(\ds 11 \, 617\) \(=\) \(\ds 4 \times 2904 + 1\)
\(\ds 11 \, 621\) \(=\) \(\ds 4 \times 2905 + 1\)
\(\ds 11 \, 633\) \(=\) \(\ds 4 \times 2908 + 1\)
\(\ds 11 \, 657\) \(=\) \(\ds 4 \times 2914 + 1\)
\(\ds 11 \, 677\) \(=\) \(\ds 4 \times 2919 + 1\)
\(\ds 11 \, 681\) \(=\) \(\ds 4 \times 2920 + 1\)
\(\ds 11 \, 689\) \(=\) \(\ds 4 \times 2922 + 1\)


It remains to be noted that:

the prime number before $11 \, 593$ is $11 \, 587$ which is $4 \times 2897 - 1$
the prime number after $11 \, 689$ is $11 \, 699$ which is $4 \times 2925 - 1$

confirming that they are not of the form $4 n + 1$.

$\blacksquare$


Historical Note

In his Curious and Interesting Numbers of $1986$, David Wells reports that Richard K. Guy discusses this in his Unsolved Problems in Number Theory of $1981$.

In Curious and Interesting Numbers, 2nd ed. of $1997$, he then attributes this result to a certain Den Haan, but gives no context.

It remains to identify who this is.


Sources

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