Aurifeuillian factorizations

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Aurifeuillean factorizations are derived from the quadratic equation

$a^{2}+b^{2}=(a+{\sqrt {2ab}}+b)(a-{\sqrt {2ab}}+b).\,$

They are named after Léon-François-Antoine Aurifeuille, who in 1873 rediscovered that

$2^{4n-2}=(2^{2n-1}+2^{n}+1)(2^{2n-1}-2^{n}+1).\,$

(see A207262). Perhaps Aurifeuille was unaware of Leonhard Euler's August 28, 1742 letter to Christian Goldbach mentioning the identity $4n^{4}+1=(2n^{2}-2n+1)(2n^{2}+2n+1)$, which is a broader class of these numbers.

Another Aurifeuillean factorization is

$3^{6n-5}+1=(3^{2n-1}+1)(3^{2n-1}+3^{n}+1)(3^{2n-1}-3^{n}+1).\,$

References

• P. H. Fuss, Correspondance math. et physique 1 (1843) p. 145
• Don Knuth, Seminumerical Algorithms, 3rd ed p392
• David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005) p. 15.