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A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.

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Fixed point theorems

Suppose that $S$ is a finite set with an odd number of elements. Then every involution $f:S\rightarrow S$ has a fixed point. Application: Every prime of the form $p=4m+1$ may be written as a sum of tw …