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The problem of simplicity of Fermat number-twins $f_{n}^{\pm}=2^{2^n}\pm3$ isstudied. The question for what $n$ numbers $f_{n}^{\pm}$ are composite isinvestigated. The factor-identities for numbers of a kind $x^2 \pm k $ arefound.

The concrete theory of numbers : Problem of simplicity of Fermat number-twins