In mathematics, Sophie Germain's identity is a polynomial factorization named after Sophie Germain stating that

Beyond its use in elementary algebra, it can also be used in number theory to factorize integers of the special form , and it frequently forms the basis of problems in mathematics competitions.[1][2][3]

History

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]

Modifying this equation by multiplying by gives

a difference of two squares, from which Germain's identity follows.[5] The inaccurate attribution of this identity to Germain was made by Leonard Eugene Dickson in his History of the Theory of Numbers, which also stated (equally inaccurately) that it could be found in a letter from Leonhard Euler to Christian Goldbach.[5][6]

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

One consequence of Germain's identity is that the numbers of the form

cannot be prime for . (For , the result is the prime number 5.) They are obviously not prime if is even, and if is odd they have a factorization given by the identity with and .[3][7] These numbers (starting with ) form the integer sequence

1, 5, 32, 145, 512, 1649, 5392, 18785, 69632, ... (sequence A001589 in the OEIS).

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

where is the fourth cyclotomic polynomial. As with the cyclotomic polynomials more generally, is an irreducible polynomial, so this factorization of infinitely many of its values cannot be extended to a factorization of as a polynomial, making this an example of an aurifeuillean factorization.[8]

Generalization

Germain's identity has been generalized to the functional equation

which by Sophie Germain's identity is satisfied by the square function.[4]

References

  1. 1 2 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
  2. 1 2 "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
  3. 1 2 3 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
  4. 1 2 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
  5. 1 2 3 Whitty, Robin, "Sophie Germain's identity" (PDF), Theorem of the day
  6. Dickson, Leonard Eugene (1919), History of the Theory of Numbers, Volume I: Divisibility and Primality, Carnegie Institute of Washington, p. 382
  7. 1 2 Bogomolny, Alexander, "Sophie Germain's identity", Cut-the-Knot, retrieved 2023-06-19
  8. Granville, Andrew; Pleasants, Peter (2006), "Aurifeuillian factorization", Mathematics of Computation, 75 (253): 497–508, doi:10.1090/S0025-5718-05-01766-7, MR 2176412
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