Arithmetic Progressions of Squares and the dir Function

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1. The eight progressions in a magic square

By the general-form theorem, every 3×3 magic square is uniquely determined by three coordinates E,x,y. With the chosen labels for the entries it has the form

\begin{pmatrix}A&B&C\\D&E&F\\G&H&J\end{pmatrix} = \begin{pmatrix} E+x&E-x+y&E-y\\ E-x-y&E&E+x+y\\ E+y&E+x-y&E-x \end{pmatrix}.

Three entries U,V,W form an arithmetic progression in the displayed order exactly when U+W=2V. In the general magic square this identity holds for exactly the following eight paths, with a path and its reversal identified:

Type	Path	Identity
Through the center	$J,E,A$	$J+A=2E$
	$H,E,B$	$H+B=2E$
	$C,E,G$	$C+G=2E$
	$D,E,F$	$D+F=2E$
Knight path	$F,A,H$	$F+H=2A$
	$D,C,H$	$D+H=2C$
	$B,G,F$	$B+F=2G$
	$B,J,D$	$B+D=2J$

In the first four sequences the center of the square is the middle term. In each of the remaining four, the move from the first term to the second and then from the second to the third is geometrically a chess knight move; these will therefore be called knight-move progressions.

Completeness of the list

After subtracting E, the nine entries have the following x,y coefficients:

\begin{pmatrix} (1,0)&(-1,1)&(0,-1)\\ (-1,-1)&(0,0)&(1,1)\\ (0,1)&(1,-1)&(-1,0) \end{pmatrix}.

Let v be the coefficient vector of the middle entry. Two distinct endpoints with vectors u and w form a progression with it exactly when w=2v−u also occurs in the table. For v=(0,0) there are four opposite pairs; for each of the four corner vectors (±1,0),(0,±1) there is exactly one pair; and for the four side vectors there are none. Hence the eight displayed progressions exhaust all identities U+W=2V among distinct entries of the general magic square.

2. Complete parametrization of three squares

Consider one of the progressions above and require all three of its terms to be rational squares. Writing them as r²,s²,t² in progression order gives

r^2+t^2=2s^2.

Thus a progression of squares is a rational point on a conic. The following theorem describes all of its points.

Theorem

All rational solutions of r²+t²=2s², and only those solutions, are given by

\begin{aligned} r&=k(-m^2+2mn+n^2),\\ s&=k(m^2+n^2),\\ t&=k(m^2+2mn-n^2), \end{aligned} \qquad k,m,n\in\mathbb Q.

Consequently, r²,s²,t² form an arithmetic progression. The signs of r,s,t do not affect its squared entries.

Forward direction

Direct expansion gives

r^2+t^2=2k^2(m^2+n^2)^2=2s^2,

so s²−r²=t²−s².

Reverse direction

Suppose r²+t²=2s². Define

u=\frac{r+t}{2},\qquad v=\frac{t-r}{2}.

Then u²+v²=s². If s+v≠0, set

m=s+v,\qquad n=u,\qquad k=\frac1{2(s+v)}.

Using u²=s²−v² gives

2kmn=u,\qquad k(m^2-n^2)=v,\qquad k(m^2+n^2)=s.

Since r=u−v and t=u+v, these are exactly the displayed formulas. In the exceptional case s+v=0, one has u=0 and v=−s; it is covered by m=0,n=1,k=s. The zero solution is covered by k=0. Hence the parametrization is surjective.

3. Primitive integral progressions

Call a nonzero integral root triple primitive when gcd(r,s,t)=1. Every nonconstant primitive triple, after choosing root signs and reversing the progression if necessary, is obtained with k=1 from coprime integers m,n of opposite parity. Every integral triple is then obtained by multiplying all roots by a common integer k.

Proof

Reducing r²+t²=2s² modulo 2 shows that r and t have the same parity. If both were even, then s²=2((r/2)²+(t/2)²) would also be even, hence s would be even as well, contradicting primitiveness. Thus r and t are odd. The square of an odd integer is 1 modulo 8, so 2s²=r²+t²≡2 (mod 8). Therefore s²≡1 (mod 4), and s is odd too. Now u=(r+t)/2 and v=(t−r)/2 are integers satisfying u²+v²=s². A common divisor would divide r=u−v and t=u+v, and then s as well, so gcd(u,v)=1. They cannot have the same parity either: even parity contradicts coprimality, while a sum of two odd squares is not a square modulo 4.

After swapping u and v, assume u is even. Then (s+v)(s−v)=u² and gcd(s+v,s−v)=2. Hence the coprime integers (s+v)/2 and (s−v)/2 have square product and are themselves squares m² and n². Thus u=2mn, v=m²−n², s=m²+n², where gcd(m,n)=1 and m,n have opposite parity. Returning to r=u−v and t=u+v gives the formulas of the preceding theorem.

The constant progression r²=s²=t² is the degenerate case mn(m²−n²)=0. It is excluded in problems requiring pairwise distinct entries.

4. The normalized difference dir(m,n)

The parameter k controls only the common scale of the roots: multiplying it by λ multiplies all three squared terms by λ². To compare progressions within one magic square, it is useful to isolate quantities that depend only on the ratio m:n. Define

f(m,n)=mn(m^2-n^2),\qquad c(m,n)=(m^2+n^2)^2,\qquad \operatorname{dir}(m,n)=\frac{f(m,n)}{c(m,n)}.

For the parametrized progression, its middle square V=s² and oriented difference Δ=s²−r²=t²−s² are

V=k^2c(m,n),\qquad \Delta=4k^2f(m,n).

If V≠0, the common scale k cancels and leaves the dimensionless ratio

\frac{\Delta}{V}=4\operatorname{dir}(m,n).

5. The eight progressions in dir coordinates

Return now to the general magic square. For each of its eight progressions, the middle term V and oriented difference Δ are linear forms in E,x,y. If its three terms are nonzero rational squares, the preceding parametrization turns the square condition into the equation Δ/V=4dir(m,n):

Path	$V$	$\Delta$	Normalized equation
$J,E,A$	$E$	$x$	$\frac{x}{E}=4\operatorname{dir}(m,n)$
$H,E,B$	$E$	$y-x$	$\frac{y-x}{E}=4\operatorname{dir}(m,n)$
$C,E,G$	$E$	$y$	$\frac{y}{E}=4\operatorname{dir}(m,n)$
$D,E,F$	$E$	$x+y$	$\frac{x+y}{E}=4\operatorname{dir}(m,n)$
$H,A,F$	$E+x$	$y$	$\frac{y}{E+x}=4\operatorname{dir}(m,n)$
$D,C,H$	$E-y$	$x$	$\frac{x}{E-y}=4\operatorname{dir}(m,n)$
$B,G,F$	$E+y$	$x$	$\frac{x}{E+y}=4\operatorname{dir}(m,n)$
$B,J,D$	$E-x$	$-y$	$\frac{-y}{E-x}=4\operatorname{dir}(m,n)$

The quotient form is convenient for comparing lines but is not needed for existence: the homogeneous equations V=k²c(m,n) and Δ=4k²f(m,n) remain valid when V=0.

6. Parameter symmetries

The dir function depends only on the ratio m:n and records the orientation of the progression. Its definition immediately gives

\begin{aligned} \operatorname{dir}(\lambda m,\lambda n)&=\operatorname{dir}(m,n),& \operatorname{dir}(n,m)&=-\operatorname{dir}(m,n),\\ \operatorname{dir}(m,-n)&=-\operatorname{dir}(m,n),& \operatorname{dir}(m+n,m-n)&=\operatorname{dir}(m,n), \end{aligned} \qquad \lambda\ne0.

Changing the sign of dir reverses the order of the squared entries, while preserving dir keeps the same normalized difference. Thus the parameters of one progression are not unique. For primitive integral triples it is convenient to choose coprime m,n of opposite parity and record order and signs separately.

7. Congruent numbers and the square class of the difference

A progression r²,s²,t² with difference Δ gives the right triangle

(t-r)^2+(t+r)^2=(2s)^2,\qquad \frac{(t-r)(t+r)}2=\Delta.

Consequently, a positive integral difference is a congruent number. For an arbitrary positive rational difference, the same construction gives a rational right triangle of area Δ; after multiplication by a rational square, one may choose an integral representative of the same square class. Since Δ=4k²f(m,n),

[\Delta]=[f(m,n)].

Two parametrized progressions can be scaled to the same nonzero difference exactly when the ratio of their f-values is a rational square. For integral parameters this class can be encoded by the squarefree part of f(m,n); this is the source of the tfmn method developed later.

8. From one progression to square configurations

The parametrization of one progression is a complete local building block: it lists every rational progression of squares and, after primitive normalization, every integral one. If a 4/9 or 5/9 square configuration contains only one nonlinear relation of this kind, one first chooses a parametrized progression and then recovers E,x,y from the remaining linear conditions.

When several progressions are required to consist of squares, their parameters must simultaneously describe shared entries and the same coordinates E,x,y. The local parametrizations are therefore supplemented by scale-compatibility equations. Equality of the square classes of the differences leads to equations between tf values, which underlie the tfmn method; other compatibility patterns produce conics, norm equations, and elliptic curves. In this way the parametrization of one progression develops into the theory of 5/9, 6/9, and 7/9 configurations.

fmn and tfmn →General theory of 4/9 and 5/9 patterns