Siragian Triangles

The square of the hypotenuse of a Siragian Triangle equals J(J+1)

Pythagoras's Rule for the calculation of the lengths of the sides of a right triangle is familiar to everyone. So well-known is this piece of elemental math that it has been repeatedly suggested that humans might construct giant replicas of the Pythagorean Theorem on Earth's surface as a means of signaling to aliens. As a warning perhaps that humans had progressed to the level of high-school geometry -- and regressed to a high-school level of adolescent and worse behavior toward our fellow beings with whom we share the Earth.

Motivated in part by analogy with the quantum mechanics of angular momentum and partly by mere curiosity, I introduced a modification to Pythagoras's Triangle which I named after my friend Saul-Paul Sirag. The Siragian Triangle differs from the Pythagorean Triangle only by the way in which you describe its hypotenuse. Phythagoras uses the symbol Z. Sirag uses the symbol J where Z and J are related by the expression:

Z^2 = J(J+1)                         (EQ 1)

The Sirag gambit does not alter the geometry of the right triangle but is merely a conventional change of variables similar to the choice to use Centigrade rather than Fahrenheit degrees to measure temperature.

One of the questions that mathematics have asked about the Pythagorean Triangle is this: How many right triangles have sides whose lengths are integers? The resulting 3 numbers for the integral lengths of two sides and the hypotenuse are called a Pythagorean Triple and can be symbolized PT(X, Y, Z).

The most familiar Pythagorean Triple is the famous 3-4-5 right triangle, but lots of integral-sided Pythogorean Triangles exist.

Similarly if one looks for integral solutions to the Siragian Triangle, one is led to the notion of a Siragian Triple ST(X, Y, J). Shortly after introducing the Sirag naming convention, I calculated a few Sirag Triples on my pocket calculator and committed a little elementary arithmetic in the privacy of my home.

Saul-Paul was immediately inspired to take the question further. "This problem reminds me of my favorite book," he said. His favorite book tells you a lot about Saul-Paul. It's Albert K. Beiler's Recreations in the Theory of Numbers: The Queen of Mathematics Entertains.

(Apparently that branch of math called Number Theory is widely recognized as "the Queen of Mathematics.")


With our meager little Sirag Triple problem in hand, opening the cover of Beiler's book was like walking through a door from a dingy alleyway into the crystal palace of the Queen of Mathematics. Our precise question was not answered in Beiler's book but we did find many closely related math problems that had caught the attention of Western mathematicians as well as Chinese, Arabs, Indians and Greeks. Saul-Paul's favorite book does indeed contain the keys to solving the problem of Sirag Triples, but for the mathematically adventurous, it holds many more treasures besides.

The Square Numbers S (N) = N^2

I learned in Beiler, for instance, about the concept of Square Numbers, which are not numbers that lack hipness, but numbers that can be formed into a square. The Square Number S(8) = 64, for example, represents the number of squares on a chess board.

The Triangle Numbers T(N) = N(N+!)/2

 Also in The Queen of Mathematics Entertains I encountered the concept of Triangular Numbers which, as their name suggests, can be arranged in a triangular array. The most familiar example of a triangular number is T(4) = 10, which pictures the way that pins are set up in a bowling alley.

But what do these cute numerical squares and triangles have to do with the Sirag Triples? Quite a lot, as it turns out. For our work in Her Majesty's service, the Queen of Mathematics Herself bestowed on Saul-Paul and me a most marvelous gift. A gift that shines. A gift that will last forever.

What we expected: Two long strings of numbers.

What we got instead: A mathematical miracle.

To understand the nature of our gift, consider the special case of the isosceles Siragian Triangle, a right triangle that possesses two equal sides, which I will call "X". The Sirag Triple then takes the isosceles form ST(X, X, J). The numerical values in this triple answer this question: What two pairs of integers X and J will satisfy (EQ 2)?

 X^2 + X^2 = J(J+1)           (EQ 2)

By trial and error, and using a pocket calculator, one can compute the first three solutions to the Isosceles Sirag Triple problem. These three solutions are ST(1, 1, 1), ST(6, 6, 8) and ST(35, 35, 49). You can check for yourself that each of these three pairs of numbers satisfies (EQ 2), hence is belonging with full benefits to the illustrious Sirag Triple Society. Here is a larger table of Isosceles Sirag Triples that cost a day and a half of my time using pocket calculator, graph paper and lots of strong coffee.

The first nine Isosceles Sirag Triples

To the casual glance, these two columns of numbers appear completely arbitrary except perhaps for the fact that they both alternate between "even" and "odd" as you progress down each column. However the apparent patternlessness of X and J conceals a remarkably simple relationship, which is the gist of Her Majesty's Mathematical Gift. Her Gift is precisely this: If we let the numbers X in the left column label the Square Numbers S(X), and the numbers J in the right column label the Triangular Numbers T(J), then anytime X and J satisfy the formula (EQ 3), then X, X and J form an Isosceles Sirag Triple.

S(X) = T(J)              (EQ 3)

In plain language, the problem of finding an Isosceles Sirag Triple is exactly the same as finding an integer that is both a Square and a Triangular Number. Dismissing the integer "1", which is a trivial case, the lowest square triangular number is 36, which is the square of 6 but also can form a triangle with base 8. For this case (EQ 3) reduces to S(6) = T(8) = 36. As you can see from the green diagram, numbers that are both Triangular and Square are exceedingly rare. Searching through the first 2 trillion numbers, I was only able to find 9 such magic numbers that satisfy this condition. (Of course, I couldn't actually test all 2 trillion digits to construct the green diagram, but used tricks to get close to the right number. Then I conducted a local search on a few likely suspects to nab the culprit.)

Given the scarcity of square triangular numbers (and their largeness, which rapidly outstrips the capacity of a pocket calculator), how does one actually find and capture more of these rare beasts? The Queen of Mathematics Entertains suggests many complicated strategies for carrying out this Quest. But there is no method shorter and more efficient than that devised recently by Armando Guarnaschelli, an amateur mathematician who lives in Argentina.

The Isosceles Sirag Triple ST(X, X, J) is not strictly speaking a triple since two of its members are alike. These numbers are in effect a Sirag Couple. Saul-Paul, playing around with Siragian Triangles,
has discovered an infinity of such triangles with unequal sides. But all such non-isosceles Siragian Triangles that Saul-Paul has so far derived conform to the type ST(X, J, J); that is, one of the triangle's sides always has the same length as the "half hypotenuse" J. Generalizing from this experience, Saul-Paul conjectures that there are no true Sirag Triples. The only Sirag Triples that exist in nature are in effect Sirag Couples which come in two varieties, namely ST(X, X, J) and ST(X, J, J). Saul-Paul's conjecture may well be true. But it could be suddenly demolished by a single counter example.

I would like to thank Saul-Paul Sirag for inspiring this effort and for putting together most of the pieces. (And thanks also for more chances to learn to spell the word "isosceles"). And deep gratitude to Her High Radiance, the Queen of Mathematics, for deigning to bless our work with such an unexpected and breathtakingly elegant outcome.

Saul-Paul Sirag in front of the Ken Kesey statue in Eugene, Oregon