000: Introduction

ii) Chernick

If u²+uv+v² = 7, then for k = 1,2,4,6,

(-u+7)^k + (u-2v+1)^k + (-3u-1)^k + (3u+2v+1)^k = (u+7)^k + (-u+2v+1)^k + (3u-1)^k + (-3u-2v+1)^k

7th powers (Piezas):

1) If x²-10y² = 9, then for k = 1,3,5,7,

1 + 5^k + (3+2y)^k + (3-2y)^k + (-3+3y)^k + (-3-3y)^k = (-2+x)^k + (-2-x)^k + (5-y)^k + (5+y)^k

2) If x²+6y² = 1, then for k = 1,2,3,5,7,

(3x+2y)^k + (-3x+7y)^k + (3x-2)^k + (3+2x)^k + (-5x-3y)^k = (-3x+2y)^k + (3x+7y)^k + (-3x-2)^k + (3-2x)^k + (5x-3y)^k

8th powers

i) Letac, Sinha

If a²+12b² = c², and 12a²+b² = d², then for k = 1,2,4,6,8,

(a+c)^k + (a-c)^k + (3b+d)^k + (3b-d)^k + (4a)^k = (3a+c)^k + (3a-c)^k + (b+d)^k + (b-d)^k + (4b)^k

ii) Wroblewski

If 17a²-33b² = 8c², and 3a²-3b² = 8d², then for k = 1,2,4,6,8,

(a+b)^k + (-a+b)^k + (a+c)^k + (a-c)^k + (b+3d)^k + (-b+3d)^k = (b+c)^k + (b-c)^k + (b+d)^k + (-b+d)^k + (2a)^k + (4d)^k

9th powers

i) Piezas

Let 283a²-475b² = 2072c², and 5491a²-47347b² = 2072d². If {p,q} = 17c ± d, {r,s} = 7c ± 5d, {t,u} = 39c ± 5d, then for k = 1,2,3,9,

(aq+br)^k + (bs-ap)^k + (bu+ap)^k + (-aq+bt)^k + (2ad-78bc)^k + (-2ad-46bc)^k =

(-aq+br)^k + (bs+ap)^k + (bu-ap)^k + (aq+bt)^k + (-2ad-78bc)^k + (2ad-46bc)^k

10th powers

i) Choudhry, Wroblewski

If 45a²-11b² = c², and -11a²+45b² = d², then for k = 1,2,4,6,8,10,

(-a-3b-c)^k + (-a-3b+c)^k + (3a-b+d)^k + (3a-b-d)^k + (2a+8b)^k + (-8a+2b)^k =

(a-3b-c)^k + (a-3b+c)^k + (3a+b+d)^k + (3a+b-d)^k + (-2a+8b)^k + (-8a-2b)^k

ii) Wroblewski, Piezas

If a²+b² = c², and a²+52b² = d², (a pair of conditions which is a concordant form). Then for k = 1,2,4,6,8,10,

(8b)^k + (5a-4b)^k + (-a-2d)^k + (a-2d)^k + (-5a-4b)^k + (-12b+4c)^k + (12b+4c)^k =

(4a+8b)^k + (3a-2d)^k + (-3a-2d)^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 6-10-8 Identity.) As another easy example, 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 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.