Primes Expressible as x^2 + n y^2 for all n from 1 to 10

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Sequence

The sequence of prime numbers which can be expressed in the form $x^2 + n y^2$ for all values of $n$ from $1$ to $10$ begins:

$1009, 1129, 1201, 1801, \ldots$

This sequence appears not to be documented on On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof


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Examples

1009

\(\ds 1009\) \(=\) \(\ds 15^2 + 1 \times 28^2 = 28^2 + 1 \times 15^2\)
\(\ds \) \(=\) \(\ds 19^2 + 2 \times 18^2\)
\(\ds \) \(=\) \(\ds 31^2 + 3 \times 4^2\)
\(\ds \) \(=\) \(\ds 15^2 + 4 \times 14^2\)
\(\ds \) \(=\) \(\ds 17^2 + 5 \times 12^2\)
\(\ds \) \(=\) \(\ds 25^2 + 6 \times 8^2\)
\(\ds \) \(=\) \(\ds \ \, 1^2 + 7 \times 12^2\)
\(\ds \) \(=\) \(\ds 19^2 + 8 \times 9^2\)
\(\ds \) \(=\) \(\ds 18^2 + 9 \times 5^2\)
\(\ds \) \(=\) \(\ds 3^2 + 10 \times 10^2\)


1129

\(\ds 1129\) \(=\) \(\ds 20^2 + 1 \times 27^2 = 27^2 + 1 \times 20^2\)
\(\ds \) \(=\) \(\ds 29^2 + 2 \times 12^2\)
\(\ds \) \(=\) \(\ds 19^2 + 3 \times 16^2\)
\(\ds \) \(=\) \(\ds 27^2 + 4 \times 10^2\)
\(\ds \) \(=\) \(\ds 2^2 + 5 \times 15^2\)
\(\ds \) \(=\) \(\ds 23^2 + 6 \times 10^2\)
\(\ds \) \(=\) \(\ds 11^2 + 7 \times 12^2\)
\(\ds \) \(=\) \(\ds 29^2 + 8 \times 6^2\)
\(\ds \) \(=\) \(\ds 20^2 + 9 \times 9^2\)
\(\ds \) \(=\) \(\ds 33^2 + 10 \times 2^2\)


1201

\(\ds 1201\) \(=\) \(\ds 24^2 + 1 \times 25^2\)
\(\ds \) \(=\) \(\ds 7^2 + 2 \times 24^2\)
\(\ds \) \(=\) \(\ds 1^2 + 3 \times 20^2\)
\(\ds \) \(=\) \(\ds 25^2 + 4 \times 12^2\)
\(\ds \) \(=\) \(\ds 34^2 + 5 \times 3^2\)
\(\ds \) \(=\) \(\ds 5^2 + 6 \times 14^2\)
\(\ds \) \(=\) \(\ds 33^2 + 7 \times 4^2\)
\(\ds \) \(=\) \(\ds 7^2 + 8 \times 12^2\)
\(\ds \) \(=\) \(\ds 25^2 + 9 \times 8^2\)
\(\ds \) \(=\) \(\ds 29^2 + 10 \times 6^2\)


1801

\(\ds 1801\) \(=\) \(\ds 35^2 + 1 \times 24^2\)
\(\ds \) \(=\) \(\ds 1^2 + 2 \times 30^2\)
\(\ds \) \(=\) \(\ds 37^2 + 3 \times 12^2\)
\(\ds \) \(=\) \(\ds 35^2 + 4 \times 12^2\)
\(\ds \) \(=\) \(\ds 26^2 + 5 \times 15^2\)
\(\ds \) \(=\) \(\ds 25^2 + 6 \times 14^2\)
\(\ds \) \(=\) \(\ds 3^2 + 7 \times 16^2\)
\(\ds \) \(=\) \(\ds 1^2 + 8 \times 15^2\)
\(\ds \) \(=\) \(\ds 35^2 + 9 \times 8^2\)
\(\ds \) \(=\) \(\ds 19^2 + 10 \times 12^2\)


Historical Note

Jekuthiel Ginsburg conjectured that:

The number $1201$ seems to be the smallest prime which can be expressed in the form $x^2 + n y^2$ for all values of $n$ from $1$ to $10$.

David Wells reported in his Curious and Interesting Numbers of $1986$ that this hypothesis was presented in Volume $8$ of Scripta Mathematica, but research is needed to track down the exact issue, date and page number.


In $1992$, Charles Ashbacher reported in Journal of Recreational Mathematics that the $1201$ is not in fact the smallest, but that $1009$ and $1129$ also have this property.

In his Curious and Interesting Numbers, 2nd ed. of $1997$, Wells provided the above update.


Sources

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