In mathematics, Sophie Germain's identity is a polynomial factorization named after Sophie Germain stating that
History edit
Although the identity has been attributed to Sophie Germain, it does not appear in her works. Instead, in her works one can find the related identity[4][5]
The identity can be proven simply by multiplying the two terms of the factorization together, and verifying that their product equals the right hand side of the equality.[7] A proof without words is also possible based on multiple applications of the Pythagorean theorem.[1]
Applications to integer factorization edit
One consequence of Germain's identity is that the numbers of the form
Many of the appearances of Sophie Germain's identity in mathematics competitions come from this corollary of it.[2][3]
Another special case of the identity with and can be used to produce the factorization
Generalization edit
Germain's identity has been generalized to the functional equation
References edit
- ^ a b Moreno, Samuel G.; García-Caballero, Esther M. (2019), "Proof without words: Sophie Germain's identity", The College Mathematics Journal, 50 (3): 197, doi:10.1080/07468342.2019.1603533, MR 3955328, S2CID 191131755
- ^ a b "CC79: Show that if is an integer greater than 1, then is not prime" (PDF), The contest corner, Crux Mathematicorum, 40 (6): 239, June 2014; originally from 1979 APICS Math Competition
- ^ a b c Engel, Arthur (1998), Problem-Solving Strategies, Problem Books in Mathematics, New York: Springer-Verlag, p. 121, doi:10.1007/b97682, ISBN 0-387-98219-1, MR 1485512
- ^ a b Łukasik, Radosław; Sikorska, Justyna; Szostok, Tomasz (2018), "On an equation of Sophie Germain", Results in Mathematics, 73 (2), Paper No. 60, doi:10.1007/s00025-018-0820-y, MR 3783549, S2CID 253591505
- ^ a b c Whitty, Robin, "Sophie Germain's identity" (PDF), Theorem of the day
- ^ Dickson, Leonard Eugene (1919), History of the Theory of Numbers, Volume I: Divisibility and Primality, Carnegie Institute of Washington, p. 382
- ^ a b Bogomolny, Alexander, "Sophie Germain's identity", Cut-the-Knot, retrieved 2023-06-19
- ^ Granville, Andrew; Pleasants, Peter (2006), "Aurifeuillian factorization", Mathematics of Computation, 75 (253): 497–508, doi:10.1090/S0025-5718-05-01766-7, MR 2176412