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Martin Sleziak
Martin Sleziak
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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 two squares. The result above is used on p.20 herehere (Wayback Machine).

Also, although not of the usual fixed point theorem form, is something I call the fixed point factor theorem. If $f:\mathbb{C}\rightarrow \mathbb{C}$ is a polynomial and $n>1$, then $f(x)-x$ is a factor of $f^n (x)-x$ for natural $n$. This has a very obvious generalisation...

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 two squares. The result above is used on p.20 here.

Also, although not of the usual fixed point theorem form, is something I call the fixed point factor theorem. If $f:\mathbb{C}\rightarrow \mathbb{C}$ is a polynomial and $n>1$, then $f(x)-x$ is a factor of $f^n (x)-x$ for natural $n$. This has a very obvious generalisation...

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 two squares. The result above is used on p.20 here (Wayback Machine).

Also, although not of the usual fixed point theorem form, is something I call the fixed point factor theorem. If $f:\mathbb{C}\rightarrow \mathbb{C}$ is a polynomial and $n>1$, then $f(x)-x$ is a factor of $f^n (x)-x$ for natural $n$. This has a very obvious generalisation...

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J.P. McCarthy
J.P. McCarthy
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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: Sorry saw this in retrospect but difficult to edit on phone. OneEvery prime of the "Proofs from the Book" uses this will add tomorrowform $p=4m+1$ may be written as a sum of two squares. The result above is used on p.20 here.

Also, although not of the usual fixed point theorem form, is something I call the fixed point factor theorem. If $f:\mathbb{C}\rightarrow \mathbb{C}$ is a polynomial and $n>1$, then $f(x)-x$ is a factor of $f^n (x)-x$ for natural $n$. This has a very obvious generalisation...

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: Sorry saw this in retrospect but difficult to edit on phone. One of the "Proofs from the Book" uses this will add tomorrow.

Also, although not of the usual fixed point theorem form, is something I call the fixed point factor theorem. If $f:\mathbb{C}\rightarrow \mathbb{C}$ is a polynomial and $n>1$, then $f(x)-x$ is a factor of $f^n (x)-x$ for natural $n$. This has a very obvious generalisation...

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 two squares. The result above is used on p.20 here.

Also, although not of the usual fixed point theorem form, is something I call the fixed point factor theorem. If $f:\mathbb{C}\rightarrow \mathbb{C}$ is a polynomial and $n>1$, then $f(x)-x$ is a factor of $f^n (x)-x$ for natural $n$. This has a very obvious generalisation...

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J.P. McCarthy
J.P. McCarthy
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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: Sorry saw this in retrospect but difficult to edit on phone. One of the "Proofs from the Book" uses this will add tomorrow.

Also, although not of the usual fixed point theorem form, is something I call the fixed point factor theorem. If $f:\mathbb{C}\rightarrow \mathbb{C}$ is a polynomial and $n>1$, then $f(x)-x$ is a factor of $f^n (x)-x$ for natural $n$. This has a very obvious generalisation...