Dedicated to Katherine Neville, author of the novel “The Eight”.1
Keywords: algebraic identities, Diophantine square identities, division algebras, quaternions, octonions, magic squares.
Contents:
I. Introduction
II. The Brahmagupta-Fibonacci Two-Square Identity
III. The Euler Four-Square Identity
IV. The Degen-Graves-Cayley Eight-Square Identity
V. Conclusion: Beyond Eight Squares
In this article, we will give an overview and discuss certain beautiful algebraic identities involving squares, and primarily of form,
(x12 + …+ xn2) (y12 + …+ yn2) = (z12 + …+ zn2)
For what n can we find such identities? Given all xi and yi as independent variables, and zi as bilinearly dependent on the xi and yi, then it turns out we can find parametric forms for n = 1, 2, 4, 8, and no other. This restriction is bound to raise some querulous eyebrows just as the restriction that the solvability in radicals of the general kth degree equation is only for k = 1, 2, 3, 4 and no higher. Why, for both, does it stop? (Digressing for a moment, note that n and k can be related as n = 2k-1. Is there a connection or is it just the law of small numbers?) Anyway, Abel and Galois gave the reason for the latter while Hurwitz dealt with the former. However, for both cases, if we relax certain conditions, then we may be able to do so for other n, though we are getting ahead of ourselves.
The importance of these identities is that they are intimately connected to certain division algebras, namely the ones for the reals, complex numbers, quaternions, and octonions, corresponding to dimension n = 1, 2, 4, 8, respectively. The interesting thing is that the mathematics in each is different. To illustrate, the reals obey a natural ordering,
a > b > … > c
and are commutative,
a + b = b + a, a b = b a
and associative,
(a + b) + c = a + (b + c), (a b) c = a (b c)
However, if we go up a division algebra of higher dimension, we lose some of these features. Complex numbers, one step up, do not have a natural ordering; they can, after all, be seen as points in the complex plane. For the quaternions, we also lose multiplicative commutativity. Finally, for the octonions, we lose associativity as well. Perhaps this can be one unrigorous way to see why there are no identities beyond n = 8: there is nothing more to lose. Hurwitz, more rigorously, would establish the theorem that bears his name why this is so.
Now that we have the preliminaries out of the way, we can go to the explicit forms of these identities. For the first n, n = 1, we have the not-so-interesting,
a2b2 = (ab)2
though this is just for starters since for other n we will need more ingenious algebraic manipulation.
This is given by,
(a2 + b2) (c2 + d2) = (ac - bd)2 + (bc + ad)2
and from this we can derive Lebesgue’s Three-Square Identity, (Henri Lebesgue, 1875-1941),
(a2 + b2 + c2 + d2)2 = (a2 + b2 - c2 - d2)2 + (2ac - 2bd)2 + (2bc + 2ad)2
The two-square identity is also known as just the Fibonacci Identity (see, for example http://mathworld.wolfram.com/FibonacciIdentity.html), named after Leonardo Fibonacci (1170-1250) better known for the Fibonacci numbers. However, in fairness to the Indian mathematician Brahmagupta (598-670 AD), it must be pointed out that he was aware of this identity, or at least a version of it, a full 500 years beforehand.
Brahmagupta did work on number theory, especially Pell equations. In 628 AD, he discovered what is known as Brahmagupta’s lemma, which is relevant to the solutions for Pell equations. For a particular application, it starts with the identity,
(b2 - na2)(d2 - nc2) = (bd + nac)2 - n(bc + ad)2
By letting n = -1, one can see that it is just the two-square identity. What the lemma implies is that if b2 - na2 = 1 and d2 - nc2 = 1, or (a,b) and (c,d) are two solutions of Pell’s equation, then from the above, we can obviously generate a third solution,
(bd + nac)2 - n(bc + ad)2 = 1
For more on Bramagupta and Pell equations, see http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pell.html#s1.
We can also use the Lebesgue three-square identity to find a 3x3 semi-magic square of squares, or an array of n x n distinct numbers such that the sum of each column and row is the same. (If the two main diagonals are included and also have the same sum, then it is a full magic square.) Our 3x3 square is given by,
(a2 + b2 - c2 - d2)2 (2bc - 2ad)2 (2ac + 2bd)2
(2ad + 2bc)2 (a2 - b2 + c2 - d2)2 (2cd - 2ab)2
(2bd - 2ac)2 (2ab + 2cd)2 (a2 - b2 - c2 + d2)2
Their common sum, of course, will be (a2 + b2 + c2 + d2)2. See Kevin Brown’s article “Orthomagic Square of Squares” at http://www.mathpages.com/home/kmath427.htm. Later, we will consider the case for a 5x5 semi-magic square of squares.
Given by,
(a2 + b2 + c2 + d2)(e2 + f2 + g2 + h2) = u12 + u22 + u32 + u42
where,
u1 = ae - bf - cg - dh
u2 = af + be + ch - dg
u3 = ag - bh + ce + df
u4 = ah + bg - cf + de
This was discovered more than a thousand years after Brahmagupta (and more than 500 years after Fibonacci who incidently also introduced Hindu-Arabic numerals to the West) by Leonhard Euler (1707-1783) in 1750. He wrote about it in a letter to Christian Goldbach (of Goldbach’s Conjecture, as if you didn’t know). The four-square identity was also used by Joseph Lagrange (1736-1813) to prove what is called Lagrange’s Four-Square Theorem, namely, that every positive integer can be expressed as the sum of four squares.
In analogy to Lebesgue’s identity, we can also find a (yours truly) Five-Square Identity,
(a2 + b2 + c2 + d2 + e2 + f2 + g2 + h2)2 = u02 + (2u1)2 + (2u2)2 + (2u3)2 + (2u4)2
where,
u0 = a2 + b2 + c2 + d2 - e2 - f2 - g2 - h2
with the other ui as previously defined. This leads to the possibility of a 5x5 semi-magic square of squares, in analogy to what Brown did. Unfortunately, the author has not been able to find the precise arrangement of expressions in this 5x5 square, nor whether if there is an unique arrangement (up to rotation and change of variables) as seems to be the case for the 3x3 square using the Lebesgue identity. It is hoped an interested reader will find the answer to this question.
Anyway, going back to the derived forms, the general identity is given by, let 2n be an integer, then,
(x12 +… x2n2)2 – (x12 +… xn2 – (xn+12 +… x2n2))2 = 4(x12 +… xn2)(xn+12 +… x2n2)
Of course, if the right hand side of the equation happens to be expressible as the sum of n squares, which is the case for n =1, 2, 4, 8, then it automatically implies a square as the sum of n+1 squares, explaining the Lebesgue three-square and our five-square identities. There will obviously be a nine-square version as well. For the case n = 1, we have,
(x12 + x22)2 - (x12 - x22)2 = 4x12x22
which is the formula for Pythagorean triples and have been known since classical times.
Finally, the last possible identity of this sort (with qualifications), is given by,
(a2 + b2 + c2 + d2 + e2 + f2 + g2 + h2) (m2 + n2 + o2 + p2 + q2 + r2 + s2 + t2) =
v12 + v22 + v32 + v42 + v52 + v62 + v72 + v82
where,
v1 = am - bn - co - dp - eq - fr - gs - ht
v2 = bm + an + do - cp + fq - er - hs + gt
v3 = cm - dn + ao + bp + gq + hr - es - ft
v4 = dm + cn - bo + ap + hq - gr + fs - et
v5 = em - fn - go - hp + aq + br + cs + dt
v6 = fm + en - ho + gp - bq + ar - ds + ct
v7 = gm + hn + eo - fp - cq + dr + as - bt
v8 = hm - gn + fo + ep - dq - cr + bs + at
(I have not seen this given explicitly anywhere in the Net and I assume nor have you. It will eventually be submitted to Mathworld and Wikipedia, of course. Knowing that there was an eight-square identity, I re-constructed it using certain heuristics. However, for those who wish to see for themselves that indeed it is true, there is available a MS Word file www.oocities.org/titus_piezas/DegenGraves2.doc with which one can cut and paste it in a computer algebra system, like the free site http://www.quickmath.com/.)
The history of this identity is a bit convoluted. It seems the first person to construct it was the Danish mathematician C. Ferdinand Degen (1766-1825) in 1818, about 70 years after Euler’s. As a side note, it can be mentioned that since Degen was the foremost Scandanavian mathematician of his time, it was to him that Niels Abel (1802-1829) as a young man of 19 submitted a “solution” to the general quintic. When Degen asked for a numerical example, Abel found his mistake, changed his mind about the solvability of the general quintic in radicals, and the rest, as they say, is history.
However, it seems Degen’s discovery went unnoticed. The next person in the story is the Irish mathematician William Rowan Hamilton (1805-1865). Hamilton, of course, is known for discovering the quaternions. The anecdote about this is almost as famous as the one involving Ramanujan and the number 1729. The story goes that Hamilton was walking on the Brougham bridge on his way to a meeting of the Royal Irish Academy when he realized how he could generalize the complex numbers and, so excited by this discovery, couldn’t resist the impulse to etch the rules i2 = j2 = k2 = ijk = -1 on the bridge.
This was October of 1843. He wrote his friend, John Thomas Graves (1806-1870), jurist and mathematician, telling him about the quaternions. Two months later, in December, Graves constructed what he first called the “octaves”, though later he would call them “octonions”. Hamilton offered to write a paper on them, but kept putting it off, until Arthur Cayley (1821-1895), having also been acquainted with the quaternions, independently found the next generalization, the eight-dimensional case, and published it in 1845. Thus, the octonions are sometimes referred to as Cayley numbers. In short, the identity was independently discovered three times.
While Wikipedia (see entry http://en.wikipedia.org/wiki/Octonions) would credit Graves with the discovery, Mathworld (see entry on http://mathworld.wolfram.com/DiophantineEquation2ndPowers.html) would credit Degen with priority. Mathworld mentions in the references “Degen, C.F. Canon Pellianus, Copenhagen, Denmark, 1817”, so it seems it really was Degen but to be fair to all parties concerned (imagine the labor involved in verifying the eight-square identity by hand!), I think it would be appropriate to name it after all three of them. Hence, the Degen-Graves-Cayley Eight-Square Identity.
V. Conclusion: Beyond Eight Squares
The natural question to ask next is: Can we find a parametric identity for the product of two sums of 16 squares expressed as the sum of 16 squares? Or,
(x12 + …+ x162) (y12 + …+ y162) = (z12 + …+ z162)
The answer, surprisingly, is yes.
While we have pointed out earlier that such identities are only for n = 1, 2, 4, 8, the phrasing of our question allows for certain interpretations. In mathematics, we may have to be excruciatingly precise when we phrase our statements. For example, if we say that there is no formula to solve the general quintic, that is not entirely accurate. There is no formula in radicals or root extractions, but if we go beyond those functions, there is certainly a formula using elliptic functions found by Charles Hermite or hypergeometric functions found by Felix Klein to solve the general quintic. Likewise, we will have to be careful with the wording of the question asked earlier. To this end, we will have to appeal to the relevant theorems established by Hurwitz and Pfister.
Theorem 1. Hurwitz’s Theorem (in 1898, by Adolf Hurwitz, 1859-1919)
Let F be a field with characteristic not 2. The sum of squares identity of the form,
(x12 + …+ xn2) (y12 + …+ yn2) = (z12 + …+ zn2)
where each zi is bilinear over xi and yi (with coefficients in F) is possible if and only if n = 1, 2, 4, 8.
However, if we relax our requirements, then we have,
Theorem 2. Pfister’s Theorem (in 1967, by Albrecht Pfister)
Let F be a field with characteristic not 2. The sum of squares identity of the form,
(x12 + …+ xn2) (y12 + …+ yn2) = (z12 + …+ zn2)
where each zi is a rational function of xi and yi (element of F(x1,…xn,y1,…yn)) is possible if and only if n is a power of 2.
For more details, see http://planetmath.org/encyclopedia/PfistersTheorem.html
Pfister’s theorem allows then that if we make one of our yi, call it y1, linearly dependent on the others, and the zi as rational functions of the xi and yi, then one can find a parametric identity for such sums for n a power of 2. Thus, for our question, if it had the unspoken assumption that one of the squares yi2 may not be arbitrary and can be dependent on the other variables, then the answer indeed is yes. And while the author has constructed one such parametrization for n = 16, we will just give a numerical example as the explicit expression for the dependent variable y1 is horribly complicated.
Let,
t1 = 22 + 32 + 52 + 62 + 72 + 92 + 112 + 122 + 132 + 172 + 192 + 202 + 232 + 252 + 272 + 352 = 4796
t2 = 12 + 42 + 152 + 162 + 182 + 212 + 242 + 252 + 282 + 292 + 322 + 362 + 372 + 392 + 2222 + 4942 = 302619
t3 = 7272 + 9372 + 10032 + 13172 + 30172 + 34242 + 38472 + 52272 + 90902 + 96632 + 97382 + 101132 + 110892 + 121732 + 185132 + 199322 = 1451360724
where the ti are sums of 16 squares. Then, t1t2 = t3 and hence is the product of two sums of 16 squares equal to the sum of 16 squares, as claimed. Small numbers were deliberately chosen for the xi and yi though they are completely arbitrary save one, namely y1 = 494 while 222 was chosen to make all numbers integral. This example is also included in the MS Word document so one can easily cut and paste it onto a computer algebra system to verify that it is true.
There is in fact an attempt to push the envelope beyond the octonions, namely a 16-dimensional algebra with elements called sedenions, though this is no longer a division algebra. See Mari Imaeda’s site “Octonions And Sedenions” at http://www.oocities.org/zerodivisor/ for more details. And it seems these sums of squares identities may even have fascinating implications for theoretical physics, as John Baez points out in his article for Week 104.
So that’s it, the sum of n squares identities for n = 1, 2, 4, 8. We have finally finished our journey through this particular region of the mathematical landscape. I do hope you enjoyed the ride.
1Author’s Note:
From the very start of writing this article, I knew exactly who I was going to dedicate it to, namely the novelist Katherine Neville who wrote the book “The Eight”. I figured that it was particularly apt. I first read that book almost half a lifetime ago, when I was 17, and was fascinated by it, by the layers upon layers of storytelling as well the topic itself, which was an adventure through history. It starts with a quest to find a very old chess service owned by Charlemagne, the Montglane. As you know, chess involves an 8x8 square, and indeed the number eight plays a major role in the novel. The book is a master alchemist’s blending of fact and fiction, of a genre similar to Umberto Eco’s “Foucault’s Pendulum” and Dan Brown’s “The Da Vinci Code”. If you like any of the two, you will like the third.
See http://www.randomhouse.com/features/magiccircle/eight.html for more details.
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© 2005
Titus Piezas III
July 16, 2005
http://www.oocities.org/titus_piezas/ramanujan.html ® Click here for an index.
tpiezasIII@uap.edu.ph ® (Remove “III” for email)
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