Timeline of Euler’s Extended Conjecture (EEC)

 

By Titus Piezas III

 

 

“Because it’s there.”- George Mallory, Mt. Everest mountaineer¹

 

 

Keywords: diophantine equations, equal sums of like powers, Euler’s extended conjecture, Fermat’s last theorem.

 

 

Dedicated to Maritess Piezas-Giraud and Jean-Charles Meyrignac (and the team at EulerNet).

 

 

I. Introduction

 

            This is an appendix to the paper “Euler’s Extended Conjecture and a^k + b^k + c^k = d^k  for k > 4” by the same author and it is recommended that one reads that before this.  EEC is a very broad conjecture² which developed from Lander et al’s 1967 paper “A Survey of Equal Sums of Like Powers” to its formalization in Ekl’s 1998 “New Results In Equal Sums Of Like Powers”.  Just like Fermat’s Last Theorem, EEC can be simply stated:

 

“The diophantine equation (k,m,n), (eq.1)

 

x₁^k + x₂^k + … + x_m^k = y₁^k + y₂^k + … + y_n^k

 

has no integral solution for k > m+n, other than the trivial case when all x_i = y_i.”

 

Since no intellectual endeavor occurs in a vacuum but rather has a genealogy behind it, it might be worthwhile to trace the development of this conjecture which has an understandable intertwining with FLT and some other problems, especially the Prouhet-Tarry-Escott Problem.  The purpose of this appendix is then to provide some sort of a concise overview of its historical development.  A summary of selected numerical results will also be given.

 

 

II. Timeline

 

(Note: Dates in red have been added or modified in the last update which is June 23, 2006.)

 

c.2600 BC  -  Egyptian scholars and pyramid architects familiar with the 3:4:5 right triangle.

c.1900 BC  -  Babylonian tablet discusses a problem with 3² + 4² = 5² as the solution.

c.  800 BC  -  Indian Baudhayana Sulbasutra gives a special case of the Pythagorean theorem.  The Sulbasutras contain many examples of particular Pythagorean triples such as (5,12,13), (12,14,20), etc.

c.  560 BC  -  Pythagoras of Samos was born.

c.  387 BC  -  Plato³ starts his Academy in Athens.  In one of his Dialogues (The Repubic, Book VIII) he alludes to, but does not explicitly state, the fact that the cube of the first perfect number (which is 6) is 3³ + 4³ + 5³ = 6³ = 216.

c.  100 BC  -  The Zhoubi suanjing (Zhou Shadow Gauge Manual) mentions the “Gougu rule”, the Chinese version of the Pythagorean theorem.

0200  -  Birth of Diophantus of Alexandria (Egypt).  Wrote about in the Arithmetica on what are now known as diophantine equations.

1202  -  Leonardo Fibonacci wrote Liber Abaci, which among other things introduced Hindu-Arabic numerals to the West eventually replacing the cumbersome Roman numerals.

1591  -  Francois Viete gives two parametric solutions to a³ + b³ + c³ = d³.  Also did work on solving cubic equations.

1630s  - Probable decade Fermat wrote his theorem on a^k + b^k = c^k for k > 2 (FLT) on the margin of Diophantus’ book.  Among these marginal notes was a proof for k = 4 using the method of “infinite descent”.

1636  -  Fermat proposes the problem of the sum of two cubes whose sum is a cube to Sainte-Croix.  Later in 1640, he submits the same problem to Frenicle de Bessy.

1657  -  de Bessy finds 1³ + 12³ = 9³ + 10³ = 1729 as an answer to another problem raised by Fermat, thereby anticipating Ramanujan and taxicab numbers.

1707  -  Leonhard Euler was born. Amongst extensive mathematical output, he would find the complete rational parametrization of (3,1,3) given in four variables and the first parametric solution to (4,2,2).

1753  -  Euler writes a letter to Goldbach claiming a “proof” of FLT for k = 3.  Later made a conjecture “…generalizing FLT” (in his own words) that “it would take at least k kth powers to sum to a kth power”, known as Euler’s Sum of Powers Conjecture (ESC).  (Euler’s proof used the properties of x²+3y² though one step contained a rectifiable error.)

1816  -  J. R. Young finds first partial parametrization of (3,1,3) in terms of quadratic forms by decomposing (2d²+4d+42)³ as the sum of three cubes of like form. (Note: Gérardin, Osborne, Ramanujan, Reznick, Womack et al found various quadratic form parametrizations of (3,1,3).  A generalization has now been found by this author.  See “Ramanujan and the Cubic Equation 3³ + 4³ + 5³ = 6³”.)

1820  -  Sophie Germain⁴ partially proves FLT for a large class of k.

1825  -  Legendre and Dirichlet, building on work done by Germain, complete the proof for k = 5.

1839  -  Lamé proves FLT for k = 7 by using the factorization of (x+y+z)⁷-(x⁷+y⁷+z⁷).

1841  -  J. Binet gives a two-variable solution to (3,1,3) by expressing the cube of ((a+3b)(a²+3b²)-1) as the sum of three cubes, though this is just a version of Euler’s general formula.  (Note: This is the same Jacques Binet of Binet’s Fibonacci Number Formula which gives the nth Fibonacci number.)

1872  -  Hermite derives Euler’s and Binet’s solution from a general property of cubic surfaces.

1878  -  E. Catalan (of Catalan’s conjecture) gives a solution to (3,1,3) using quartic forms. (This conjecture states that 8 and 9 are the only consecutive powers.  Preda Mihailescu’s proof in 2002 is widely accepted as finally settling this problem.)

1880  -  A. Desboves finds a complex two-parameter solution to (5,2,2) involving (x²+2y²)Ö-1.

1898  -  E. Fauquembergue finds the first parametrization of (4,1,5) by decomposing (4x⁴+y⁴)⁴ as five fourth powers.

1904  -  C.B. Haldeman solves (4,1,5) in terms of quadratic forms.  These two identities are the same as given by Ramanujan.

1906  -  The Wolfskehl Prize⁵ is announced, offering 100,000 Marks for a proof for FLT.  (Note: This would later be awarded to Andrew Wiles in 1997.  In 2005, Wiles would be awarded the recently-established Shaw Prize worth US $1 million for proving FLT.)

1911  -  R. Norrie by “…a series of special assumptions” finds the first biquadrates for (4,1,4).

1912  -  Hayashi gives a solution to (4,2,2) by solving the related algebraic form 3u⁴ + v⁴ = w².

1915  -  R. Carmichael (of Carmichael numbers) using the algebraic form x³+y³+z³-3xyz derives Euler’s solution.  (Note: The aforementioned numbers satisfy a different FLT, namely, Fermat’s Little Theorem.)

1934  -  S. Sastry finds a 2-parameter solution to (5,1,5) by decomposing (u⁵+75v⁵)⁵ as five fifth powers.

1934  -  Subba Rao gives the first parametric solution to (6,3,3).

1939  -  Moessner finds first (5,3,3).

1942  -  Patterson finds second (4,1,4) thirty years after the first. (Note: About eighty-seven (4,1,4) are now known with sum e⁴ and e < 35,000, though it is not known whether there is a parametric solution.)

1943  -  Swinnerton-Dyer (at the tender age of 16) publishes his first paper⁶ on the arithmetic of surfaces, a parametrization of (4,2,2).

1948  -  Morgan Ward in his “Euler’s Problem On Sums Of Three Fourth Powers” proves that (4,1,3) or a⁴ + b⁴ + c⁴ = d⁴ has no solution for d < 10000 by “…using congruential restraints⁷ and some hand computing.”

1951  -  Moessner finds a two-parameter solution to (5,3,3).

1952  -  Swinnerton-Dyer gives a parametrization to (5,3,3) though it results in polynomials of a high degree.

1955  -  Yutaka Taniyama (1927-1958) with Goro Shimura proposes a fascinating connection between modular forms and elliptic curves, later known as the Taniyama-Shimura conjecture.  If true, the statement implies the validity of FLT.  (Note: A few years later Taniyama would tragically commit suicide.  See Footnote on Wolfskehl Prize.)

1958  -  J. Leech finds the next six (4,1,4) using the EDSAC 2 computer.  (John Leech is better known for the 24-dimensional Leech lattice.  The ancestor of the computer he used, EDSAC, was used by Birch and Swinnerton-Dyer to gather numerical evidence for what is known as, of course, the Birch and Swinnerton-Dyer Conjecture.)

1967  -  Lander, Parkin, and Selfridge find no solution to (4,1,3) for d < 220,000 using Ward’s constraints and the state-of-the-art CDC 6600. (Note:  The CDC 6600, operating 1966-77, was the first to be designated as a “supercomputer”.  R. Frye using a faster descendant would eventually find the smallest solution at d = 422481.)

1967  -  Lander and Parkin in their pivotal paper “A Counterexample to Euler’s Sum of Powers Conjecture”, as well as in “A Survey of Equal Sums of Like Powers” with Selfridge, prove ESC wrong by serendipitously finding (5,1,4).  The computer program was meant to find (5,1,5) but yielded a zero term for one solution!

1968  -  Lander, using a geometric approach, gives a general method to find more solutions to (4,2,2) using polynomials of degree 7, 13, 19, etc.  Also finds one with three parameters defining a quartic surface giving (k,3,3) for k = 1,5.

1969  -  Brudno solves (4,2,2) using different polynomials of degree 13, 19, 31.

1970  -  Brudno gives a method to find an infinite number of solutions to (6,3,3).

1973  -  Demjanenko in his paper “L. Euler’s Conjecture” gives a parametrization of r⁴ + s⁴ + t⁴ = 1 as a pencil of conics.

1981  -  Bremner applies Lander’s geometric approach to both fifth and sixth powers.

1983  -  Zajta gives more parametrizations for (4,2,2) using modified Pythagorean triples.

1987  -  Elkies, building on Demjanenko’s work, gives a method to generate an infinite number of (4,1,3) and finds the first counter-examples to ESC for degree k = 4.

1988  -  R. Frye, using Elkies’ result and the Connection Machine, finds the second and smallest (4,1,3) after more than 100 machine hours spread over several weeks.

1992  -  Delorme gives more parametric solutions to (6,3,3).

1993  -  Richard Guy in the chapter on Euler’s conjecture in his book “Unsolved Problems in Number Theory” asks for solutions to (7,1,7) and (7,4,4), among others.  Also gives a bound on (5,2,2) as having no solution with a common sum less than 1.02 x 10²⁶, implying any solution must have one term on each side > 138,000.

1993  -  Andrew Wiles presents a proof for a special case of the Taniyama-Shimura Conjecture.  Unfortunately, a flaw is found in the proof involving a defective Euler system.

1995  -  With the help of former student Richard Taylor, Wiles gives a corrected version of the proof.  After more than 350 years, FLT is finally proven true.

1996  -  Robert Scher and Ed Seidl find the first (5,2,3) using certain congruential constraints⁸ and a week of computer time on a 72-node Intel Paragon.

1996  -  R. Ekl using a modified AVL tree algorithm and a 50 MHz HP Unix finds first, as well as smallest, (7,4,4) in 53 minutes.  (Note: About forty are now known.)

1997  -  Chen Shuwen starts his “Equal Sums of Like Powers” site on August focusing on the related Prouhet-Tarry-Escott Problem.  Now hosted at http://euler.free.fr/eslp/eslp.htm.

1998  -  Ekl in “New Results In Equal Sums Of Like Powers” formalizes EEC by extending Euler’s original conjecture with insights from Lander et al’s paper.  Improves the bound quoted in Guy for (5,2,2) as 4.01 x 10³⁰, thus any solution should have at least two terms > 1,149,000.

1998  -  Ekl later finds first (9,5,5) while MacLeod finds third (4,1,3).

1999  -  Jean-Charles Meyrignac starts EulerNet, “Computing Minimal Equal Sums Of Like Powers” on Jan 3, 1999 at http://euler.free.fr/.  Three months later, team member M. Dodrill finds first (7,1,7).

1999  -  A. Choudhry finds a quartic polynomial in three variables such that whenever it assumes non-trivial square values, one can solve a multi-grade (k,3,3) for k = 1,2,6.

2000  -  Scott Chase, independently of EulerNet, finds first (8,1,8) and (8,3,5). Maurice Blondot finds second (7,1,7).

2000  -  EulerNet extends the bound of (6,1,6) as having a sum z⁶ with z > 200,000 done by searching (6,1,7) and allowing one term to be zero.  The search has been stopped since 2001. L

2000  -  A. Choudhry using a cubic equation in five variables finds examples of multi-grade (k,4,4) for k = 1,3,7 when the cubic has a rational root.

2001  -  D. J. Bernstein, using Ward’s constraints but no longer with hand computation, finds all seven (4,1,3) with d < 21,000,000.  (An eighth found by MacLeod is beyond this range.)

2002  -  Nuutti Kuosa finds third (7,1,7).  Jaroslaw Wroblewski finds second (9,5,5).

2004  -  Wroblewski finds six more (9,5,5), one of which is multi-grade for k = 1,3,9.

2004  -  Jim Frye finds second (5,1,4).

2005  -  ???

 

 

(Author’s Note:  I’ll be regularly updating this timeline.  If you have any comments or suggestions, especially for inclusion (see rules in the next section) feel free to email at tpiezas(at)gmail.com.)

 

 

III. Numerical Results

 

            Since a lot is now known to make this exclusive we will restrict our list (as well as the timeline above) into three categories corresponding to the questions asked in the concluding remarks of Lander et al: Type I (k > m+n), Type II (k = m+n), and Type III (k < m+n), with further restrictions for the third.  Some remarks:

 

a)      If parametric solutions are known, only the smallest instance will be listed.

b)      Only the first few known solutions will be included, especially if they are getting too large. (After all, only one example suffices to prove that the equation is solvable.)

c)      Exponents k will be below a certain bound, though will be extended as needed.

 

A. Type I (k>m+n)

 

              (No examples known)

 

B. Type II (k=m+n)

 

(3,1,2):   (No solutions by Fermat’s Last Theorem)

(4,1,3):   2,682,440⁴ + 15,365,639⁴ + 18,796,760⁴ = 20,615,673⁴  (N. Elkies, 1986)

               95,800⁴ + 217,519⁴ + 414,560⁴ = 422,481⁴ (smallest, Roger Frye, 1988)

               673,865⁴ + 1,390,400⁴ + 2,767,624⁴ = 2,813,001⁴ (Allan MacLeod, 1998)

               1,705,575⁴ + 5,507,880⁴ + 8,332,208⁴ = 8,707,481⁴ (D. J. Bernstein, 2001)

(4,2,2):   59⁴ + 158⁴ = 133⁴ + 134⁴  (Euler, parametric, c.1750)

(5,1,4):   27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵  (Lander, Parkin, 1967)

               55⁵ + 3183⁵ + 28969⁵ + 85282⁵ = 85359⁵ (Jim Frye, 2004)

(5,2,3):   5027⁵ + 6237⁵ + 14068⁵ = 220⁵ + 14132⁵ (Scher, Seidl, 1996)

(6,1,5):

(6,2,4):

(6,3,3):   3⁶ + 19⁶ + 22⁶ = 10⁶ + 15⁶ + 23⁶  (Rao, 1934, parametric, valid k = 2,6)

(7,1,6):

(7,2,5):

(7,3,4):

(8,1,7):

(8,2,6):

(8,3,5):   81⁸ + 539⁸ + 966⁸ = 158⁸ + 310⁸ + 481⁸ + 725⁸ + 954⁸  (Chase, 2000)

(8,4,4):

(9,1,8):

(9,2,7):

(9,3,6):

(9,4,5):

(10,1,9):

(10,2,8):

(10,3,7):

(10,4,6):

(10,5,5):

 

C. Type III (k<m+n)

 

This will be restricted to the most difficult type, the “Eulerian” (k,1,k) and an aesthetic type, the “balanced equation” (2n-1,n,n). 

 

Eulerian solutions (k,1,k):

 

(2,1,2):  3² + 4² = 5² (antiquity, parametric)

(3,1,3):  3³ + 4³ + 5³ = 6³ (antiquity, Plato, parametric)

(4,1,4):  30⁴ + 120⁴ + 272⁴ + 315⁴ = 353⁴  (Norrie, 1911)

              240⁴ + 340⁴ + 430⁴ + 599⁴ = 651⁴  (Patterson, 1942)

              435⁴ + 710⁴ + 1384⁴ + 2420⁴ = 2487⁴  (Leech, 1958)

(5,1,5):   7⁵ + 43⁵ + 57⁵ + 80⁵ + 100⁵ = 107⁵  (Sastry, 1934, parametric, third smallest)

              19⁵ + 43⁵ + 46⁵ + 47⁵ + 67⁵ = 72⁵ (Lander, Parkin, 1967, smallest)

(6,1,6):

(7,1,7):  127⁷ +258⁷ + 266⁷ + 413⁷ + 430⁷ + 439⁷ + 525⁷ = 568⁷  (Dodrill, 1999)

              91⁷ + 148⁷ + 158⁷ + 255⁷ + 258⁷ + 309⁷ + 625⁷ = 626⁷ (Maurice Blondot, 2000)

              177⁷ + 848⁷ + 1619⁷ + 1931⁷ + 2429⁷ + 2665⁷ + 2854⁷ = 3157⁷ (Nuutti Kuosa, 2002)

(8,1,8):   90⁸ + 223⁸ + 478⁸ + 524⁸ + 748⁸ + 1088⁸ + 1190⁸ + 1324⁸ = 1409⁸  (Chase, 2000)

(9,1,9):

(10,1,10):

(11,1,11):

 

(Note: Lander et al may have established that if (6,1,6) has a sum z⁶ for some integral z, then z > 38300.)

 

Balanced equations (2n-1,n,n):

 

(3,2,2):  1³ + 12³ = 9³ + 10³  (Frenicle de Bessy, 1657; Ramanujan, 1919, parametric)

(5,3,3):  39⁵ + 92⁵ + 100⁵ = 49⁵ + 75⁵ + 107⁵  (Moessner, 1939, parametric found in 1951)

  24⁵ + 28⁵ + 67⁵ = 3⁵ + 54⁵ + 62⁵  (Lander, Parkin, Selfridge, 1967, smallest, valid k = 1,5)

(7,4,4):  10⁷ + 14⁷ + 123⁷ + 149⁷ = 15⁷ + 90⁷ + 129⁷ + 146⁷  (Ekl, 1996)

              23⁷ + 105⁷ + 150⁷ + 194⁷ = 38⁷ + 132⁷ + 152⁷ + 192⁷  (Ekl, 1996)

              19⁷ + 52⁷ + 112⁷ + 354⁷ = 35⁷ + 46⁷ + 281⁷ + 343⁷  (Michael Lau, 1999)

              184⁷ + 443⁷ + 556⁷ + 698⁷ = 230⁷ + 353⁷ + 625⁷ + 673⁷  (Nuutti Kuosa, 1999)

(9,5,5):   26⁹ + 30⁹ + 91⁹ + 101⁹ + 192⁹ = 12⁹ + 17⁹ + 116⁹ + 175⁹ + 180⁹  (Ekl, 1997)

               2⁹ + 228⁹ + 305⁹ + 819⁹ + 1735⁹ = 762⁹ + 1062⁹ + 1067⁹ + 1564⁹ + 1634⁹  (Wroblewski, 2002)

               51^k + 253^k + 412^k + 600^k + 624^k = 100^k + 187^k + 429^k + 603^k + 621^k  (Wroblewski, 2004, k = 1,3,9)

               63⁹ + 234⁹ + 435⁹ + 546⁹ + 891⁹ = 25⁹ + 172⁹ + 671⁹ + 789⁹ + 842⁹  (Wroblewski, 2004)

(11,6,6):

 

(Note: As about forty (7,4,4) and eight (9,5,5) are now known, the question is whether if these also have a parametric solution.  It applies in general to (2n,n,n) and (2n-1,n,n).)

 

 

IV. Concluding Remarks

 

            Equal sums of like powers share one feature in common with prime factorizations of extremely large integers: they are easy to do in one direction.  Given two humongous primes p and q of several hundred digits, it is trivial to multiply them together.  But given only the product pq, it is extremely hard to recover the two primes, the movie Sneakers⁹ notwithstanding.  In a similar manner, to find, say, six sixth powers equal to a sixth power is hard (and what more if we go to the range of a hundred hundredth powers) but once they are found it is easy to verify the equality.

 

            This in fact is one of its appeals, namely, that it can be verified by virtually anyone.  The first thing I did when I read in MathTrek last year about the second (5,1,4) found, 55⁵ + 3183⁵ + 28969⁵ + 85282⁵ = 85359⁵, was to click on the small calculator that is a feature in all computers and see for myself that it was true.  No surprise, but it was.  So when those six sixth powers will be found (or, if)…

 

            Generally, the higher the exponent, the more terms required and the more difficult it would be to find solutions to equations (k,m,n) of a particular form such as the Eulerian (k,1,k), or for (k,1,k-1) or (m+n,m,n), if they can be found at all.  If they can’t, then the question shifts to Why not?  What would be prohibiting solutions to such diophantine equations?  And where would these questions lead to?  It might be interesting to speculate that since FLT lead to a relationship between elliptic curves and modular forms, then EEC (or at least special cases of it since it is so broad) might be for a connection between certain algebraic surfaces or hypersurfaces and a still unspecified (or perhaps, even undiscovered) mathematical object.  Purely speculative, but you never know.

 

 

--End--

 

Footnotes:

 

1. When asked in 1924 why he wanted to climb Mt. Everest, George Mallory gave the famous answer, “Because it’s there.”  Difficult mathematical problems can be just like a mountain.

 

2. A stronger version of EEC was formalized by Meyrignac as the Lander-Parkin-Selfridge (LPS) Conjecture which states that given eq.1 “…for every k > 3, m+n >= k”.  There are two ways one can find a counter-example to this: (a) the existence of a k > m+n, OR (b) proof of the non-existence of an m+n >= k with k > 3.  The explicit formulation of EEC only forbids (a).

 

3. The positive integer 6³ = 216 is sometimes known as Plato’s number. See http://mathworld.wolfram.com/PlatosNumbers.html as well as http://www.saliu.com/philosophy/Republic_VIII.html.

 

4. Sophie Germain is the only woman in this timeline so far.  Lagrange (whose lectures notes she studied) and Gauss (whom she corresponded with) first knew her as a “M. LeBlanc”, the former knowing her identity only after he was curious about the student who submitted a particularly insightful paper and Gauss after the French invaded his hometown and Germain, via a highly-placed family friend, intervened for his safety. A class of primes are named after her: if both p and 2p+1 are prime, then p is a Sophie Germain prime.

 

5. The story behind the Wolfskehl Prize is almost as enigmatic as the one for FLT.  The first version I read about was in Simon Singh’s book about how a despondent and suicidal Paul Wolfskehl (who was an industrialist and amateur mathematician) visited his library a few hours before the appointed time for his suicide and became so engrossed in FLT and what he thought was a proof that he forgot his, er, appointment. Though he later realized his proof was invalid, in gratitude for this “salvation” he would set up this Prize.  Subsequently, I would learn that an article in the AMS Notices by Dr. Klaus Barner would give two other alternative theories. If indeed the first version is true, it is such a contrast to what later happened to Taniyama.

 

6. He wrote this paper under the name P.S. Dyer.  Of course, he is now known better as Sir Peter Swinnerton-Dyer, of the Birch and Swinnerton-Dyer Conjecture which happens to be one of the Clay Mathematics Institute’s “Millenium Problems”.  This paper, as well as a second in the other name, was in the references in Lander’s “Geometric Aspects…” though Lander didn’t seem to have noticed that the two were the same person.  It took some Internet sleuthing before I found out that indeed it was the case.

 

7. One of Ward’s congruential constraints was that given a⁴ + b⁴ + c⁴ = d⁴, then either c+d or -c+d must be divisible by 2¹⁰.  For the one found by Frye, note that 217,519 + 422,481 = 2¹⁰5⁴.

 

8. A congruential contraint used by Scher and Seidl is that if (5,0,5) has a solution then the sum of either one or two pairs of terms is divisible by 5.  The one found by Lander et al has 27 + 133 = 160 and 84 + (-144) = 60.

 

9. Sneakers, for those who somehow missed it, is one of the few movies that dealt with a mathematical subject (others were Good Will Hunting and Cube).  It starred Robert Redford (et al) as a former hacker who steals a “black box” from a mathematician.  It turned out to be a device able to find the prime factorization of extremely large numbers and hence was the ultimate code breaker.

 

 

For more on EEC, see “Euler’s Extended Conjecture and a^k + b^k + c^k = d^k  for k > 4”.

 

 

© Titus Piezas III

Oct. 21, 2005

http://www.geocities.com/titus_piezas/ramanujan.html ¬ Click here for an index

 

 

References:

 

1. Bernstein, D. J., “Enumeration Solutions To p(a)+q(b)=r(c)+s(d)”, Math of Comp, Vol 70, 2001, 389-394.
2. Bremner, A. “A Geometric Approach to Equal Sums of Fifth Powers”, J. Number Theory, V. 13, 337-354, 1981.
3. Bremner, A., “A Geometric Approach to Equal Sums of Sixth Powers”, Proc. London Math Soc., Vol 43, 544-581, 1981.
4. Brudno, Simcha, “Some New Results on Equal Sums of Like Powers”, Math of Comp, Vol 25, 1969, 877-880.
5. Brudno, Simcha, “On Generating Infinitely Many Solutions of the Diophantine Equation A⁶+B⁶+C⁶=D⁶+E⁶+F⁶”, Math of Comp, Vol 24, 1970, 453-454.
6. Choudhry, A., “On Equal Sums of Sixth Powers”, Rocky Mount. Jour. of Math, Vol 30, No. 3, 2000.
7. Choudhry, A., “Equal Sums of Seventh Powers”, Rocky Mount. Jour. of Math, Vol 30, No. 3, 2000.
8. Delorme, J., “On The Diophantine Equation x₁⁶+ x₂⁶+ x₃⁶= x₄⁶+ x₅⁶+ x₆⁶”, Math. Of Comp., Vol 59, Oct 1992, pp. 703-715.
9. Demjanenko, V., “L. Euler’s Conjecture”, Acta Arithmetica, 25 (1973-4), 127-135.
10. Dickson, L., History of the Theory of Numbers, Vol 2, Carnegie Institute of Washington, 1919. Reprinted by Chelsea Publishing, NY, 1971.
11. Ekl, R., “Equal Sums of Four Seventh Powers”, Math. Of Computation, Vol. 65, 1996.
12. Ekl, R., “New Results In Equal Sums Of Like Powers”, Math. Of Computation, Vol 67, 1998.
13. Elkies, N, “On A⁴ + B⁴ + C⁴ = D⁴”, Math of Comp, Vol 51, Oct 1988, pp. 828-838.
14. Hayashi, T., “On the Diophantine Equation x⁴+y⁴ = z⁴+ t⁴”, Tohoku Math J., Vol 1 (1912), 143-145.
15. Lander, L., and Parkin, T., “A Counter-Example to Euler’s Sum of Powers Conjecture”, Math of Comp, Vol 21, 1967, 101-103.
16. Lander, L., Parkin, T., and Selfridge, J., “A Survey of Equal Sums of Like Powers”, Math. of Comp, Vol 21, 1967, 446-459.
17. Lander, L., “Geometric Aspects of Diophantine Equations Involving Equal Sums of Like Powers”, Amer. Math. Monthly, Vol 75, 1968, pp.1061-1073.
18. Leech, J., “On A⁴ + B⁴ + C⁴ + D⁴ = E⁴”, Proc. Cambridge Philos. Soc., Vol 54, 1958, 554-555.
19. McLean, Joseph, “Equations Involving Sums of Powers”, http://www.glasgowg43.freeserve.co.uk/sumintro.htm
20. Merignac, Jean-Charles, Computing Minimal Equal Sums of Like Powers, http://euler.free.fr/
21. Moessner, A., “Due Sistemi Diofantei”, Boll. Un. Mat. Ital., (3), 6 (1951) 117-118.
22. O’Connor, J. and Robertson, E., http://www-groups.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics/
23. Peterson, I, “Euler’s Sum of Powers”, http://www.sciencenews.org/articles/20040925/mathtrek.asp
24. Sastry, S., “On Sums of Powers”, Jour. London Math. Soc., Vol 9, 1934, 242-246.
25. Shuwen, Chen, “Equal Sums of Like Powers”, http://euler.free.fr/eslp/eslp.htm
26. Singh, S., The Wolfskehl Prize, http://www.simonsingh.com/Wolfskehl_Prize.html.
27. Subba Rao, K., “On Sums of Sixth Powers”, J. London Math. Soc., Vol 9, 1934, 172-173.
28. Swinnerton-Dyer, H.P (as P.S. Dyer), “A Solution of A⁴+B⁴=C⁴+D⁴, J. London Math Soc., 18, 1943, 2-4.
29. Swinnerton-Dyer, H.P., “A Solution of A⁵+B⁵+C⁵=D⁵+E⁵+F⁵”, Proc. Cambridge Philos. Soc. 48, 1952, 516-518.
30. Ward, Morgan, “Euler’s Problem On Sums Of Three Fourth Powers”, Duke Math J., Vol. 15, 1948, pp. 827-837.
31. Womack, Tom, “Equal Sums of Powers” (dissertation), www.tom.womack.net/maths/dissert.pdf
32. Zajta, A., “Solutions of the Diophantine Equation A⁴ + B⁴ = C⁴ + D⁴”, Math of Comp, Vol 41, 1983, 635-659.
33. et al.
34. http://oll.libertyfund.org/Texts/Plato0204/Dialogues/HTMLs/0131-03_Pt01_Intros.html