Abstract
We deal with the following functional equation
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
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.
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http://www.theoremoftheday.org/Binomial/GermainId/TotDGermainIdentity.pdf
https://topologicalmusings.wordpress.com/2008/02/20/sophie-germain-identity/
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Łukasik, R., Sikorska, J. & Szostok, T. On an Equation of Sophie Germain. Results Math 73, 60 (2018). https://doi.org/10.1007/s00025-018-0820-y
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DOI: https://doi.org/10.1007/s00025-018-0820-y
Keywords
- Sophie Germain identity
- quadratic functional equation
- biadditive and symmetric functions
- functional equations on integral domains