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How to Improve a Math History Assignment - Euler and a Problem from Number Theory

Christopher Goff (University of the Pacific)

Euler and a Problem from Number Theory

Leonhard Euler (1707-1783) was perhaps the most prolific mathematician in history.  His contributions to mathematics and physics are many, too many to list here.  Euler worked at the St. Petersburg Academy from 1727 to 1741, at which point he went to work at the Academy of Science in Berlin at the invitation of Frederick the Great.  After working there for 25 years, Euler moved back to St. Petersburg, where he worked until he died.  Euler wrote so much that it took the St. Petersburg Academy journal almost 50 years after his death to publish the remainder of his papers.  (Note:  For an interesting take on how Euler's later works may have served his pedagogical aims, see Ed Sandifer’s column How Euler Did It from February 2010.)

The paper to be discussed here, "De tribus pluribusve numeris inveniendis, quorum summa sit quadratum, quadratorum vero summa biquadratum" (Eneström index number E763), or “On finding three or more numbers, the sum of which is a square, while the sum of the squares is a fourth power,” was one of Euler's posthumously published works.  Although written in 1780, it first appeared in print in 1824 in the Memoirs of the St. Petersburg Academy.  All direct quotations from E763 that appear below are from my own translation, available at the MAA Euler Archive.

Leonhard Euler (1707-1783)
(Source: MacTutor Archive)

Title of Euler's paper in its original Latin, as it appeared in the Mémoires d l'Académie
impériale des sciences de St. Pétersbourg (Memoirs of the St. Petersburg Imperial
Academy of Sciences) in 1824.  (Source:  Digital image contributed to Biodiversity
Heritage Library
by Natural History Museum Library, London)

Euler's E763 refers to and extends an earlier problem of Pierre de Fermat, namely, to find the two legs of a right triangle whose hypotenuse is a square and whose legs add up to a square.  Curiously, Fermat came across this problem while reading Claude Bachet’s 1621 translation from Greek to Latin of the Arithmetica of Diophantus (c. 250 CE), who posed the original problem. Mathematically, then, the original problem was to find two numbers whose sum is a square and whose squares add up to a fourth power.  First, Euler mentioned that Fermat and Lagrange had studied the problem, and then described his own early difficulties with trying to resolve the question of finding three integers with the same properties.  This led him back to Fermat’s original problem, the smallest solution of which involves integers greater than a billion.  Despite this initial setback, Euler was ultimately able to invent a general approach to finding solutions to problems of this type.

Euler’s Technique

In a nutshell, Euler exploited the Euclidean formulas for generating Pythagorean triples.  He first stated the original problem: to find \(x\) and \(y\) so that \(x + y\) is a square and \(x^2 + y^2\) a fourth power.  In his own words (using \(xx\) for \(x^2\), etc.) [§.5.]:

Let us begin with the latter condition.  At first, indeed, the formula \(xx + yy\) shall be rendered as a square, by placing \(x = aa-bb\) and \(y = 2ab,\) for then \(xx + yy = {(aa + bb)}^2.\)  In addition, then, this formula \(aa + bb\) should be a square, which happens in the same way by setting \(a = pp-qq\) and \(b = 2pq\):  from here, it follows that \(xx + yy = {(pp + qq)}^4,\) and thus the latter condition has now been fully satisfied.

Thus Euler used nested formulas for generating Pythagorean triples to guarantee that, regardless of his choice of \(p\) and \(q,\) \(x^2 + y^2\) would be a fourth power.  So then, he needed \(x + y\) to be a square.  After a bit of effort, he determined that \(p = 1469\) and \(q = 84,\) leading him to a solution [§.7.]:

These numbers are therefore

\(x =  4,565,486,027,761\)

\(y =  1,061,652,293,520\)

which are the same that Fermat, and others after him, found.  The sum of them is the square of the number \(2,372,159,\) while the sum of the squares is the fourth power of the number \(2,165,017.\)

For the three-variable version of the problem, Euler adapted the Euclidean formulas [§.8.]:

Let us begin again from the sum of squares, which is first rendered as a square, by placing \(x = aa + bb-cc; \) \(y = 2ac; \) \(z = 2bc; \) which thus will become \(xx + yy + zz = {(aa + bb + cc)}^2; \) whereby thus \(aa + bb + cc\) ought to be made a square again, which will be done in a similar way by putting \(a = pp + qq-rr;\) \(b = 2pr;\) \(c = 2qr;\) for thus is obtained \(xx + yy + zz = {(pp + qq + rr)}^4;\) so that the latter condition is now fulfilled.

Next, Euler found a relationship between \(p,\) \(q,\) and \(r,\) namely that \(p=r+{\frac{3}{2}}q,\) for which the corresponding values of \(x,\) \(y,\) and \(z\) would add up to a square.  His first example was \(x = 409,\) \(y = 152,\) and \(z = 64\) (corresponding to \(q = 2,\) \(r = 1,\) and thus \(p = 4\)), the sum of which is \(625 = 25^2\) and the sum of whose squares is \(194,481 = 441^2= 21^4.\) 

Euler handled the case of finding four numbers in much the same way, and found a similar relationship among his lowest-level variables, namely: \(p=s+{\frac{3}{2}}r-q,\) which would guarantee a solution.  For the case of five variables, Euler found that  \(p=t+{\frac{3}{2}}s-r-q\) would lead to a solution.  He then stated [§.23.]:

and thus for the case of six numbers it will be found that, \(p=u+{\frac{3}{2}}t-s-r-q,\) and so forth, from which the general question, proposed for any number of numbers, must now be considered completely solved.

As was the case with Turing, Euler went even further, but since his continued explorations are beyond the scope of this article, I will move on to the student work.  Interested readers may consult the original paper

Student Work: “Engaged” Group

In her paper, the student explained what Euler did, step by step from the beginning, and she even filled in some of the missing computational details behind his explanation of the three-number (\(x\)-\(y\)-\(z)\) case.  She then pointed out his general pattern and proceeded to use it to find six numbers with the same properties.  As her lowest-level variables, she selected \(u = t = r = 2,\) and \(s = q = 1,\) which meant \(p = 1.\)  This led her to the following six numbers:  \(97,\) \(112,\) \(64,\) \(64,\) \(128,\) and \(64,\) whose sum is \(529 = 23^2,\) and whose squares add up to \(50,625 = 225^2 = 15^4.\)

This student did what I expected.  She found a primary document, explained its content, and extended the mathematics beyond its original scope.  However, when selecting a topic, this student had struggled.  After a few unsuccessful attempts at finding primary source material that interested her personally, I eventually steered her toward E763, largely because I had just finished a draft of its translation from Latin to English.  The choice was convenient for her and for me, but I doubt that it piqued her curiosity.  Rather, she was doing what was asked, even if she was perhaps personally uninterested in the material.  Because this student’s experience was so different from the student who wrote about Turing and morphogenesis, I decided to look in more detail at the assignment itself.

Christopher Goff (University of the Pacific), "How to Improve a Math History Assignment - Euler and a Problem from Number Theory," Loci (July 2014)