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Equation1286.md

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An equation with non-trivial linear models

Using the linear magma model x ◇ y = ax + by, we can produce a model of this law that contradicts Equation 3 when a+b ≠ 1, b^2=-a/(a^2+1) and b=(2a^2+1)/(a^5+a^3). This can be satisfied by taking a to be a root of a^5+2a^4+3a^3+3a^2+a+1. For instance, working in ℤ/pℤ, one can take (p,a,b) = (11,1,7) or (p,a,b) = (7,6,2) (the latter is known to be the smallest model). See this discussion.