Gurevič Identities

The Gurevič Identities are a series of identities such that, for every sound, finite set of axioms of the theory of non-zero Natural Numbers, one identity cannot be proved. They are given by:

(Ax+Bnx)2x⋅(Cn2x+Dn2x)x=(A2x+Bn2x)x⋅(Cnx+Dnx)2x⋅superscriptsuperscript𝐴𝑥superscriptsubscript𝐵𝑛𝑥superscript2𝑥superscriptsuperscriptsubscript𝐶𝑛superscript2𝑥superscriptsubscript𝐷𝑛superscript2𝑥𝑥⋅superscriptsuperscript𝐴superscript2𝑥superscriptsubscript𝐵𝑛superscript2𝑥𝑥superscriptsuperscriptsubscript𝐶𝑛𝑥superscriptsubscript𝐷𝑛𝑥superscript2𝑥(A^{x}+B_{n}^{x})^{2^{x}}\cdot(C_{n}^{2^{x}}+D_{n}^{2^{x}})^{x}=(A^{2^{x}}+B_{% n}^{2^{x}})^{x}\cdot(C_{n}^{x}+D_{n}^{x})^{2^{x}}

where n ≥ 3 𝑛 3 n\geq 3 and odd, and

A𝐴\displaystyle A =1+xabsent1𝑥\displaystyle=1+x
Bnsubscript𝐵𝑛\displaystyle B_{n} =∑i=0n−1xiabsentsuperscriptsubscript𝑖0𝑛1superscript𝑥𝑖\displaystyle=\sum_{i=0}^{n-1}x^{i}
Cnsubscript𝐶𝑛\displaystyle C_{n} =1+xnabsent1superscript𝑥𝑛\displaystyle=1+x^{n}
Dnsubscript𝐷𝑛\displaystyle D_{n} =∑i=0n−1x2⁢iabsentsuperscriptsubscript𝑖0𝑛1superscript𝑥2𝑖\displaystyle=\sum_{i=0}^{n-1}x^{2i}

These identities can be generalized to get the Generalized Gurevič Identities.