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One can show (see below for a sketch of a proof) that every odd prime number $p$ can be written in exactly $(p+1)/2$ different ways as $$p=a\cdot b+c\cdot d$$ with $a,b,c,d\in\mathbb N$ satisfying $\max(c,d)<\min(a,b)$.

Example for $p=23$: \begin{matrix} 1\cdot 23+0\cdot 0 & 23\cdot 1+0\cdot 0 \\ 2\cdot 11+1\cdot 1 & 11\cdot 2+1\cdot 1 \\ 3\cdot 7+1\cdot 2 & 3\cdot 7+2\cdot 1 \\ 7\cdot 3+1\cdot 2 & 7\cdot 3+2\cdot 1 \\ 4\cdot 5+1\cdot 3 & 4\cdot 5+3\cdot 1 \\ 5\cdot 4+1\cdot 3 & 5\cdot 4+3\cdot 1 \end{matrix}

Klein's Vierergruppe $\mathbb V$ acts on all such quadruplets $(a,b,c,d)$ by permuting the first two, permuting the last two, or permuting both the first two and the last two elements. So we get an easy proof that every prime $p$ congruent to $1$ modulo $4$ must be a sum of squares: $(p+1)/2$ is then odd and the only fixed points under the action of $\mathbb V$ are of the form $(a,a,c,c)$.

Does somebody know a reference for this proof? It looks a bit like Zagier's proof which also uses a parity argument for a set acted upon by involutions.

Motivation: This is in fact a variation of Arithmetic properties of positively reduced $2\times 2$-matrices .

Sketch of proof Given a solution $(a,b,c,d)$ we consider $u=(a,c),\ v=(-d,b)$. We associate to $(a,b,c,d)$ the sublattice $\Lambda=\mathbb Zu+\mathbb Zv$ of index $p$ in $\mathbb Z^2$. Suppose now $cd>0$ and consider the eight open cones of $\mathbb R^2$ defined by the complement of the four lines defined by $xy(x^2-y^2)=0$. We colour these open cones alternatingly black and white. The four vectors $\pm u,\pm v$ are contained in different black cones (colouring the first cone above the halfline $(\mathbb R_{>0},0)$ in black).

We say that a sublattice $\Lambda$ of finite index in $\mathbb Z^2$ has a monochromatic basis if there exists a basis $b_1,b_2$ of $\Lambda=\mathbb Z b_1+\mathbb Z b_2$ such that all four elements $\pm b_1,\pm b_2$ belong to different open cones of the same colour.

(Not every lattice has a monochromatic basis but many do.)

We claim that all monochromatic bases of a lattice (having a monochromatic basis) are of the same colour: If $b_1,b_2$ is a black monochromatic basis and $w_1,w_2$ is a white monochromatic basis, then $w_1,w_2$ belong to two open adjacent cones of $\mathbb R^2\setminus(\mathbb Rb_1\cup \mathbb R b_2)$ which is impossible by the following small Lemma:

Lemma: If $f_1,f_2$ and $g_1,g_2$ are two bases of a lattice $\Lambda=\mathbb Z f_1+\mathbb Z f_2=\mathbb Z g_1+\mathbb Z g_2$ such that $\{\pm f_1,\pm f_2\}$ and $\{\pm g_1,\pm g_2\}$ do not intersect, then $g_1,g_2$ or $g_1,-g_2$ are contained in a common connected component of $\mathbb R^2\setminus(\mathbb R f_1\cup \mathbb R f_2)$. (Otherwise we have up to sign changes and exchanges of indices $f_1=\alpha b_1+\beta b_2$ and $f_2=\gamma b_1-\delta b_2$ with $\alpha,\beta,\gamma,\delta$ strictly positive integers. This implies that $b_1$ belongs to the convex hull of $(0,0),f_1,f_2$ which is a contradiction.)

Set now $\Lambda_\mu=\{(x,y)\in\mathbb Z,\ \vert\ x+\mu y\equiv 0\pmod p\}$. If $\mu\in \{2,\ldots,p-2\}$, then $\Lambda_\mu$ contains no elements of the form $(\pm m,0),(0,\pm m),(\pm m,\pm m)$ with $m$ in $\{1,\ldots,p-1\}$. This implies that every open black or white cone contains a point with coordinates of absolute value at most $p-1$. A reduction algorithm implies the existence of a monochromatic basis. (Start with two arbitrary non-zero elements $e_1,e_2$ of $\Lambda$ having coordinates of absolute value smaller than $p$ which belong to two different consecutive black cones. If the interior of the convex hull spanned by $\pm e_1,\pm e_2$ contains a non-zero element in a black cone, we can replace $e_1$ or $e_2$ and decrease the area of the convex hull spanned by $\pm e_1,\pm e_2$. If the interior contains no non-zero elements of $\Lambda$ in black cones, we get either a monochromatic basis or the convex hull contains at least four lattice points in four distinct white cones and we switch the working colour to white.)

Moreover, since $\Lambda_\mu$ and $\Lambda_{p-\mu}$ differ by a horizontal reflection, they have monochromatic bases of different colours. Retaining only lattices with black monochromatic bases, We get $(p-2-2+1)/2=(p-3)/2$ such lattices with black monochromatic bases.

Monochromatic bases of a lattice $\Lambda_\mu$ are not unique but in finite number. It remains to show that exactly one black monochromatic basis of a lattice $\Lambda_\mu$ has the form $u=(a,c),v=(-d,b)$ as required for a solution of $p=ab+cd$ (with $\min(a,b)>\max(c,d)$ and $0\leq c,d$). We call such a basis a reduced monochromatic basis. First observe that every lattice with a black monochromatic basis $b_1,b_2$ (where we suppose $b_1\in\mathbb N^2$ and $b_2$ in $(-\mathbb N)\times\mathbb N$) has a reduced black monochromatic basis: Replace $b_1$ by $b_1-kb_2$ with $k$ maximal for monochromaticity. Replace then $b_2$ by $b_2+sb_1$ with $s$ maximal for monochromaticity. The resulting black monochromatic basis is reduced.

Observing that the two lattices $\mathbb Z(p,0)+\mathbb Z(1,\pm 1)$ contain no vectors $u,v$ (associated to a solution $(a,b,c,d)$ such that ...) and adding the two trivial solutions $(p,1,0,0),(1,p,0,0)$ (corresponding to the lattices $\mathbb Z(p,0)+\mathbb Z(0,1)$ and $\mathbb Z(1,0)+\mathbb Z(0,p)$) we get a total number of at least $(p-3)/2+2=(p-1)/2$ solutions and we are done after showing that every lattice with a black monochromatic basis contains only one reduced monochromatic basis (also using the fact that sublattices of prime index $p$ are in one-to-one correspondence with points of the projective line over $\mathbb F_p$).

Supose now that $u=(a,c),v=(-d,b)$ is a reduced black basis and let $u'=(a',c'),v'=(-d',b')$ be a second reduced black basis giving rise to two distinct solutions $(a,b,c,d)$ and $(a',b',c',d')$. Since $u$ and $v$ determine each other uniquely in a reduced black basis, we can assume that $u'\not=u$ and $v'\not=v$.

The two vectors $u',v'$ are thus contained in the four open cones defined by $\mathbb R^2\setminus(\mathbb R u\cup\mathbb R v)$.

The lemma used previously shows that they can not belong to two adjacent cones of $\mathbb R^2\setminus(\mathbb R u\cup\mathbb R v)$.

We suppose now that $u',v'$ belong to $\mathcal C\cup (-\mathcal C)$ for $\mathcal C$ a cone (onnected component) of $\mathbb R^2\setminus(\mathbb R u\cup \mathbb R v)$. If $u'$ and $v'$ belong to two opposite cones, we exchange the basis $u,v$ with the basis $u',v'$. We can now assume that both vectors $u'$ and $v'$ belong to the open cone $(0,+\infty)u+ (0,+\infty)v$ spanned by $u$ and $v$. We have thus $u'=\alpha u+\beta v$ and $v'=\gamma u+\delta v$ with $\alpha,\beta,\gamma,\delta$ strictly positive integers. Reducedness of the black monochromatic basis $u,v$ implies that $v+u$ is either white or belongs to the black cone $\mathcal C_u$ containing $u$. If $u+v$ is white, we get a contradiction by observing that it is contained in the convex hull of $(0,0),u',v'$ (which contains no other points of $\Lambda$). The point $u+v$ is thus in the black cone $\mathcal C_u$ containing $u$. Geometric considerations imply now $\delta\geq 3$ and the impossible inequalities $$p\geq (3 b+c)a/2>(3ab+ac)/2>ab+a(b+c)/2>ab+cd=p\ .$$

Indeed, since $u$ and $u+v$ belong both to $\mathcal C_u$, the line $u+\mathbb Rv$ of slope $<-1$ intersects the white cone separating $\mathcal C_u$ from the black cone $\mathcal C_v$ containing $v$ in a segment containing at least one lattice-point of $\Lambda$. This implies $\delta\geq 3$ and the second coordinate of $v'$ is thus at least equal to $3b+c$. On the other hand, the first coordinate of $u'$ has to be at least equal to the first coordinate of the intersection $u+\mathbb Rv \cap \mathbb R(1,1)$ which is $\geq a/2$ (since $v$ has slope $<-1$).

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    $\begingroup$ I learned recently that, although I am sure I have only ever heard Viergruppe, it is apparently Vierergruppe. \\ The not-very-difficult starting fact is too difficult for me. Does it go too far afield to sketch a proof? $\endgroup$
    – LSpice
    Oct 5 at 21:47
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    $\begingroup$ As explained by Mathologer, Zagier's proof can be thought of as boiling down to showing that for primes $p \equiv 1\pmod{4}$, the equation $p = x^2 + 4yz$ has an odd number of positive integer solutions $(x,y,z)$. So it seems to be not quite the same proof. You might look at Christian Elsholtz's paper, A combinatorial approach to sums of two squares and related problems, to see if your proof is implicitly in there somewhere. $\endgroup$ Oct 6 at 1:43
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    $\begingroup$ @LSpice : Viergruppe corrected to Vierergruppe. I have added a slightly sketchy proof. $\endgroup$ Oct 6 at 6:56
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    $\begingroup$ Regardless of whether new or not, this is a real neat proof, Roland. $\endgroup$ Oct 6 at 10:16
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    $\begingroup$ A quite recent proof by A. David Christopher (A partition-theoretic proof of Fermat's two squares theorem. Discrete Math. 339 (2016), no. 4, 1410–1411) also splits p into such a sum of products of two factors, $ a_1 f_1+ a_2 f_2$. $\endgroup$ Oct 6 at 15:34

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