It’s funny how sometimes a simple but useful mathematical trick can go unnoticed for a long time, yet be obvious when it is pointed out. Here is a good example.
A few days ago I posted here about ,Diophantine Quadruples which are sets of four different non-zero numbers (rational or integer) such that the product of any two is one less than a square. This problem of finding them has been around since Diophantus and has been looked at by some of the greatest classical number theorists including Fermat and Euler, as well as 20th Century specialists of number theory such as Baker and Davenport. In the last few years the literature on the subject has been steadily growing. I myself have been thinking about the problem on and off since I heard about it as a teenager 30 years ago. Well, we all seem to have missed an easy trick for finding them.
Actually this method applies to a generalization of the problem which is to find sets of four distinct positive integers such that the product of any two is less than a square by a fixed number n. These quadruples are said to have the property D(n). For n = 1 there is an infinite set of them called regular quadruples that can be constructed recursively. There are probably no irregular examples but that has not been settled yet. When n is not a square the problem is more difficult with only a few examples known for each n. For the state of the art you should consult the references listed by Anrej Dujella.
So let’s suppose we have a quadruple (a,b,c,d) with property D(n)
ab = x2+n
ac = y2+n
bc = z2+n
ad = r2+n
bd = s2+n
cd = t2+n
Now form a symmetric matrix with the quadruple down the diagonal and the square roots as the correspondng off diagonal elements
( a x y r ) D = ( x b z s ) ( y z c t ) ( r s t d )
Next form the Adjoint matrix. For non-singular matrices this is the inverse times the determinant.
Adj(D) = D-1 det(D)
The elements of this matrix can also be constructed from the 3×3 minor determinants of D and this works even when D is singular. It is therefore a matrix of integers just like D, and it is symmetric.
( k u v w ) Adj(D) = ( u l e f ) ( v e m g ) ( w f g n )
The 2×2 minors of an adjoint 4×4 matrix are given by the opposing 2×2 minors from the original matrix times its determinant. e.g
kl – u2 = (cd – t2) det(D)
This works for all the minors, but we only need to look at the principal minors. The result should now be obvious. Since (cd – t2) = -n, it follows that kl – u2 = -n det(D). With the same being true for each minor, it follows that the quadruple (k,l,m,n) is also a Diophantine quadruple with the property D(n det(D)), although this could fail in specific cases if two of the numbers are the same or are not positive.
In most cases this means that if you give me a Diophantine quadruple, I can give you another one just by inverting its matrix. In fact I can give you (potentially) eight of them! This is because I can reverse the signs of the square roots and there are eight distinct cases that will usually give eight different quadruples. Obviously this process can be repeated by flipping signs in the adjoint and inverting again.
By the way, I only discovered this trick because I first realised some years ago that regular quadruples can be arranged on a 2x2x2 hypermatrix with zero hyperdeterminant. Then recently some papers appeared that showed that the minors of a symmetric matrix also give a singular hypermatrix . Putting the two together made me look at the matrix generated from a quadruple and its inverse.
It may not be a revolutionly result in mathematics but it does show how it is easy to miss simple tricks even for problems that have been around for a long time, and it is just possible that this trick could provide a new attack on some of the still unsolved problems concenring Diophantine quadruples.