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On an Equation of Sophie Germain

  • Radosław  Łukasik
  • Justyna Sikorska
  • Tomasz Szostok
Open Access
Article
  • 137 Downloads

Abstract

We deal with the following functional equation
$$\begin{aligned} f(x)^2+4f(y)^2 = \big ( f(x+y)+f(y) \big ) \big ( f(x-y)+f(y) \big ) \end{aligned}$$
which is motivated by the well known Sophie Germain identity. Some connections as well as some differences between this equation and the quadratic functional equation
$$\begin{aligned} f(x+y)+f(x-y)=2f(x)+2f(y) \end{aligned}$$
are exhibited. In particular, the solutions of the quadratic functional equation are expressed in the language of biadditive and symmetric functions, while the solutions of the Sophie Germain functional equation are of the form: the square of an additive function multiplied by some constant. Our main theorem is valid for functions taking values in a unique factorization domain. We present also an example which shows that our main result does not hold in each integral domain.

Keywords

Sophie Germain identity quadratic functional equation biadditive and symmetric functions functional equations on integral domains 

Mathematics Subject Classification

39B05 

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of SilesiaKatowicePoland

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