“Mathematics, rightly viewed, possesses not only truth, but supreme beauty.”   – Bertrand Russell

A Collection of Algebraic Identities is a 200+ page book I wrote that was inspired, in part, by the unusual Ramanujan 6-10-8 Identity,

 

 

which is true if ad = bc. Its form and use of certain high exponents is certainly intriguing (a quality that is characteristic of most of Ramanujan’s work).  Later we will see that this can be generalized.  Trying to look for other examples of unusual algebraic identities in the Internet, or at least collections of which, I couldn’t find anything substantial.  So I decided to upload my work.  It covers an eclectic range and includes basic identities, rather mundane ones such as the Bramagupta-Fibonacci Two-Square Identity,

 

(a²+b²)(c²+d²)  = (ac+bd)² + (ad-bc)²

 

to more unusual and exotic examples, including one of the most complete collection of identities for Equal Sums of Like Powers for fifth and higher powers. 

II. Why?

 

“Politics is for the moment, but an equation is forever.” – Albert Einstein

 

The argument can be given that identities are simply tautologies A = A and hence do not impart anything new. In fact, that is not the case. One can algebraically deform each side of the equation, separate it into components, give it structure, to such an extent that it is no longer immediately apparent the two sides are equal to each other.  (One need only look at the 6-10-8 Identity.)  As another easy example, the structure of the Two-Square Identity given above implies that the product of the sums of two squares is itself the sum of two squares, and this in fact belongs to a finite family of identities important to division algebras. Still another is Hirschhorn’s Odd-Even Identity which proves that the sum of four distinct odd squares is the sum of four distinct even ones.  And so on.

 

The great thing about mathematical truths, other than their unreasonable effectiveness in the sciences as discussed in a famous essay by the physicist E. Wigner is that, in contrast to physical truths (such as the statement, “There are still polar bears.”), they are eternally and universally true. Thus, if you discover a mathematical truth like a new algebraic identity (okay, so I’m a Platonist), you are looking at an object which is true forever.  For me, that is reason enough to look for them. Admittedly though, there can be more prosaic reasons.

 

Another nice aspect to algebraic identities is that they usually need only elementary mathematics and, as such, is accessible to a broad audience. (However, one may sometimes come across concepts like elliptic curves, resultants, Pell equations, quadratic forms, etc.)

 

With the advent of computer algebra systems (CAS) like Mathematica, Maple, and others, it’s now more convenient to find – and verify – algebraic identities.  Of course, it would take some ingenuity to solve the more complicated cases, especially those involving powers higher than the fourth. However, for anyone interested in finding new identities, I highly suggest taking a look at what CAS have to offer. (Mathematica's Home Edition is offered at just $295, or one can go to the free but limited site www.quickmath.com.)

III. Submissions

 

a)      a new algebraic identity

b)      a new solution to an old form

c)      a new form that needs a solution

d)      questions

e)      corrections

f)        comments

feel free to email at tpiezas@gmail.com and help enlarge and improve this database of algebraic identities.  When submitting a new algebraic identity, pls include, if possible, the general principle by which it was found since it would be nice to know if it can be generalized.  Quite a lot of the material in this book is independent work by this author.  If a particular result turns out to be a rediscovery, pls email me so I can give proper credit to that person (kindly also include a citation to the relevant book or paper).  This is just a first draft – there is still a lot of work to be done – but one has to start somewhere.