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Title: On an equation of Sophie Germain
Authors: Łukasik, Radosław
Sikorska, Justyna
Szostok, Tomasz
Keywords: Sophie Germain identity; quadratic functional equation; biadditive and symmetric functions; functional equations on integral domains
Issue Date: 2018
Citation: Results in Mathematics, Vol. 73, no. 2 (2018), art. no. 60
Abstract: We deal with the following functional equation (Formula Presented.) 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 (Formula Presented.) 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.
DOI: 10.1007/s00025-018-0820-y
ISSN: 1422-6383
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