I. Introduction (Wordpress.com blog is here.) “Mathematics, rightly viewed, possesses not only truth, but supreme beauty.” – Bertrand Russell A Collection of Algebraic Identities is a 300+ page book I wrote that was inspired, in part, by the unusual Ramanujan 6108 Identity,
64[(a+b+c)^{6}+(b+c+d)^{6}+(ad)^{6}(c+d+a)^{6}(d+a+b)^{6}(bc)^{6}] [(a+b+c)^{10}+(b+c+d)^{10}+(ad)^{10}(c+d+a)^{10}(d+a+b)^{10}(bc)^{10}] =
45[(a+b+c)^{8}+(b+c+d)^{8}+(ad)^{8}(c+d+a)^{8}(d+a+b)^{8}(bc)^{8}]^{2}
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 it will be seen that this can be generalized. One I found that also uses extremely high exponents is, define, then the two are neatly related as, x^{5} – 10ax^{3 }+ 45a^{2}x – a^{2} = 0 The quintic in fact is the Brioschi quintic form, which the general quintic can be reduced into, while {a,x} involve polynomial invariants of the icosahedron. 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 the basics, like the BramaguptaFibonacci TwoSquare Identity (BTS),
(a^{2}+b^{2})(c^{2}+d^{2}) = (ac+bd)^{2} + (adbc)^{2} its generalizations like Vladimir Arnold's perfect forms, an example being,
(2a^{2}+ab+3b^{2})(2c^{2}+cd+3d^{2}) = 2x^{2}+xy+3y^{2}, where {x,y} = {a(c+d)+b(c+2d), a(c+d)+b(cd)}
hence the product of two {2,1,3} quadratic forms (which has discriminant d = 23) is of like form, analogous to the BTS which is just the case {1,0,1}. Then it goes to more unusual and exotic examples, including one of the most complete collection of identities for Equal Sums of Like Powers for 4th and higher powers. The ancient Greeks only knew the Pythagorean theorem, (a^{2}b^{2})^{2} + (2ab)^{2} = (a^{2}+b^{2})^{2} but, more than 2000 years later, we know much more. Who knows what we'll discover in the next thousand years. A few examples are:
i) Vieta
(a^{4}2ab^{3})^{3} + (a^{3}b+b^{4})^{3} = (b^{4}2a^{3}b)^{3} + (ab^{3}+a^{4})^{3} ii) Ramanujan
(3a^{2}+5ab5b^{2})^{3} + (4a^{2}4ab+6b^{2})^{3} + (5a^{2}5ab3b^{2})^{3} = (6a^{2}4ab+4b^{2})^{3} (which gives rise to the cute equation 3^{3 }+ 4^{3 }+ 5^{3} = 6^{3})
i) Fauquembergue
(a^{4}4b^{4})^{4} + 2(2a^{3}b)^{4} + 2(4ab^{3})^{4} = (a^{4}+4b^{4})^{4}
ii) Gerardin
(a+3a^{2}2a^{3}+a^{5}+a^{7})^{4} + (1+a^{2}2a^{4}3a^{5}+a^{6})^{4} = (a3a^{2}2a^{3}+a^{5}+a^{7})^{4} + (1+a^{2}2a^{4}+3a^{5}+a^{6})^{4}
i) Choudhry
(aa^{3}2a^{5}+a^{9})^{5} + (1+a^{2}2a^{6}+2a^{7}+a^{8})^{5} + (2a^{3}+2a^{4}2a^{7})^{5} = (a+3a^{3}2a^{5}+a^{9})^{5} + (1+a^{2}2a^{6}2a^{7}+a^{8})^{5} + (2a^{3}+2a^{4}+2a^{7})^{5}
(Differences between LHS and RHS are highlighted in blue.)
ii) Chernick
If 11a^{2}+4b^{2} = 9c^{2}, and 4a^{2}+b^{2} = 9d^{2}, then for k = 1,2,3,5,
(bd)^{k} + (bd)^{k} + (2d)^{k} = (a+2d)^{k} + (a+2d)^{k} + (c2d)^{k} + (c2d)^{k}
i) Brudno, Delorme: For k = 2,6,
(n^{4}n^{3}5n^{2}+8n+8)^{k} + ((n^{3}+7n2)(n+2))^{k} + (9n^{2}+6n+12)^{k} = ((n^{2}n+3)(n+2)^{2})^{k} + (4n^{3}5n^{2}8n+8)^{k} + (n^{4}+n^{2}+14n+4)^{k} ii) Chernick
If u^{2}+uv+v^{2} = 7, then for k = 1,2,4,6,
(u+7)^{k} + (u2v+1)^{k} + (3u1)^{k} + (3u+2v+1)^{k} = (u+7)^{k} + (u+2v+1)^{k} + (3u1)^{k} + (3u2v+1)^{k}
7th powers (Piezas):
1) If x^{2}10y^{2} = 9, then for k = 1,3,5,7,
1 + 5^{k} + (3+2y)^{k} + (32y)^{k} + (3+3y)^{k} + (33y)^{k} = (2+x)^{k} + (2x)^{k} + (5y)^{k} + (5+y)^{k}
2) If x^{2}+6y^{2} = 1, then for k = 1,2,3,5,7,
(3x+2y)^{k} + (3x+7y)^{k} + (3x2)^{k} + (3+2x)^{k} + (5x3y)^{k} = (3x+2y)^{k} + (3x+7y)^{k} + (3x2)^{k} + (32x)^{k} + (5x3y)^{k}
i) Letac, Sinha
If a^{2}+12b^{2} = c^{2}, and 12a^{2}+b^{2} = d^{2}, then for k = 1,2,4,6,8,
(a+c)^{k} + (ac)^{k} + (3b+d)^{k} + (3bd)^{k} + (4a)^{k} = (3a+c)^{k} + (3ac)^{k} + (b+d)^{k} + (bd)^{k} + (4b)^{k}
ii) Wroblewski
If 17a^{2}33b^{2} = 8c^{2}, and 3a^{2}3b^{2} = 8d^{2}, then for k = 1,2,4,6,8,
(a+b)^{k} + (a+b)^{k} + (a+c)^{k} + (ac)^{k} + (b+3d)^{k} + (b+3d)^{k} = (b+c)^{k} + (bc)^{k} + (b+d)^{k} + (b+d)^{k} + (2a)^{k} + (4d)^{k}
i) Piezas
Let 283a^{2}475b^{2} = 2072c^{2}, and 5491a^{2}47347b^{2} = 2072d^{2}. If {p,q} = 17c ± d, {r,s} = 7c ± 5d, {t,u} = 39c ± 5d, then for k = 1,2,3,9,
(aq+br)^{k} + (bsap)^{k} + (bu+ap)^{k} + (aq+bt)^{k} + (2ad78bc)^{k} + (2ad46bc)^{k} = (aq+br)^{k} + (bs+ap)^{k} + (buap)^{k} + (aq+bt)^{k} + (2ad78bc)^{k} + (2ad46bc)^{k}
i) Choudhry, Wroblewski
If 45a^{2}11b^{2} = c^{2}, and 11a^{2}+45b^{2} = d^{2}, then for k = 1,2,4,6,8,10,
(a3bc)^{k }+ (a3b+c)^{k }+ (3ab+d)^{k }+ (3abd)^{k }+ (2a+8b)^{k }+ (8a+2b)^{k} = (a3bc)^{k }+ (a3b+c)^{k }+ (3a+b+d)^{k }+ (3a+bd)^{k }+ (2a+8b)^{k }+ (8a2b)^{k}
ii) Wroblewski, Piezas
If a^{2}+b^{2} = c^{2}, and a^{2}+52b^{2} = d^{2}, (a pair of conditions which is a concordant form). Then for k = 1,2,4,6,8,10,
(8b)^{k} + (5a4b)^{k} + (a2d)^{k} + (a2d)^{k} + (5a4b)^{k} + (12b+4c)^{k} + (12b+4c)^{k} = (4a+8b)^{k} + (3a2d)^{k} + (3a2d)^{k} + (4a+8b)^{k} + (16b)^{k} + (a+4c)^{k} + (a+4c)^{k}
As the equations are homogenous, it does not matter if the variables are to be solved either in the integers or the rationals, though trivial values should be avoided. And since two quadratics define an elliptic curve, all these examples have an infinite number of rational solutions. These are just a few given in the book, but not much is really known for higher powers. For example, no identity is yet known for k = 1,3,5,7,9 with a minimal number of terms. It is hoped that this collection will spur others to find more; some identities have a simplicity, clarity, and beauty all their own. 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 6108 Identity.) As another easy example, the TwoSquare 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 OddEven Identity which proves that the sum of four distinct odd squares is the sum of four distinct even ones. And the one involving the Brioschi quintic has a deeper meaning which is related to the symmetries of the icosahedron. (Some of the identities in this book may have connections to other mathematical objects.)
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), then 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
If you have:
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 something new, kindly 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. P.S. A video based on Jean Giono's short story, “The Man Who Planted Trees”. It is an inspiring story of finding purpose, and how things can change when one has vision and perseverance.
