Euler's Sum of Like Powers Conjecture history and related problems

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Euler’s Sum of Like Powers Conjecture
history and related problems
2011 Michigan MAA Section Meeting
Western Michigan University
David Murphy
Hillsdale College
May 7, 2011
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Diophantus and Diophantine equations
Little is known about Diophantus of Alexandria, sometimes called
the “father of algebra.”
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Diophantus and Diophantine equations
Little is known about Diophantus of Alexandria, sometimes called
the “father of algebra.”
◮ He lived between 150 B.C. and 350 A.D.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Diophantus and Diophantine equations
Little is known about Diophantus of Alexandria, sometimes called
the “father of algebra.”
◮ He lived between 150 B.C. and 350 A.D.
◮ “...his boyhood lasted 1/6th of his life; he married after 1/7th
more; his beard grew after 1/12th more, and his son was born
5 years later; the son lived to half his father’s age, and the
father died 4 years after the son” – 500 A.D.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Diophantus and Diophantine equations
Little is known about Diophantus of Alexandria, sometimes called
the “father of algebra.”
◮ He lived between 150 B.C. and 350 A.D.
◮ “...his boyhood lasted 1/6th of his life; he married after 1/7th
more; his beard grew after 1/12th more, and his son was born
5 years later; the son lived to half his father’s age, and the
father died 4 years after the son” – 500 A.D.
◮ He wrote Arithmetica, a collection of 189 problems in number
theory that require positive rational solutions.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Diophantus and Diophantine equations
Little is known about Diophantus of Alexandria, sometimes called
the “father of algebra.”
◮ He lived between 150 B.C. and 350 A.D.
◮ “...his boyhood lasted 1/6th of his life; he married after 1/7th
more; his beard grew after 1/12th more, and his son was born
5 years later; the son lived to half his father’s age, and the
father died 4 years after the son” – 500 A.D.
◮ He wrote Arithmetica, a collection of 189 problems in number
theory that require positive rational solutions.
From Book II, Problem 8 of his Arithmetica:
To divide a given square into two squares.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Pierre de Fermat (1601–1665)
“It is impossible for a cube to be the sum of two cubes,
a fourth power to be the sum of two fourth powers, or
in general for any number that is a power greater than
the second to be the sum of two like powers. I have
discovered a truly marvelous demonstration of this
proposition that this margin is too narrow to contain.”
— Fermat, 1630’s
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Pierre de Fermat (1601–1665)
“It is impossible for a cube to be the sum of two cubes,
a fourth power to be the sum of two fourth powers, or
in general for any number that is a power greater than
the second to be the sum of two like powers. I have
discovered a truly marvelous demonstration of this
proposition that this margin is too narrow to contain.”
— Fermat, 1630’s
Fermat’s Last Theorem
The equation
x n + y n = z n
has no solutions in positive whole numbers whenever n ≥ 3.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
The end of a 350-year-old problem
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Leonhard Euler (1707–1783)
“It has seemed to many Geometers that this
theorem [Fermat’s Last ‘Theorem’] may be
generalized. Just as there do not exist two
cubes whose sum or difference is a cube,
it is certain that it is impossible to exhibit three
biquadrates whose sum is a biquadrate, but that at least four
biquadrates are needed if their sum is to be a biquadrate, although
no one has been able up to the present to assign four such
biquadrates. In the same manner it would seem impossible to
exhibit four fifth powers whose sum is a fifth power, and similarly
for higher powers.” — Euler, 1769
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Euler’s Conjecture
The equation
x k 1 + x k 2 + · · · + x k n = y k .
has no solutions in positive whole numbers whenever n < k.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Euler’s Conjecture
The equation
x k 1 + x k 2 + · · · + x k n = y k .
has no solutions in positive whole numbers whenever n < k.
Notice that if Euler’s Conjecture is true, then Fermat’s Last
Theorem is the special case where n = 2 and k > 2.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Euler’s Conjecture
The equation
x k 1 + x k 2 + · · · + x k n = y k .
has no solutions in positive whole numbers whenever n < k.
Notice that if Euler’s Conjecture is true, then Fermat’s Last
Theorem is the special case where n = 2 and k > 2.
For nearly 200 years, no significant progress was made on Euler’s
Conjecture.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Euler was wrong!
In 1966, L. J. Lander and T. R. Parkin found four fifth powers that
sum to a fifth power:
27 5 + 84 5 + 110 5 + 133 5 = 144 5 .
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Euler was wrong!
In 1966, L. J. Lander and T. R. Parkin found four fifth powers that
sum to a fifth power:
27 5 + 84 5 + 110 5 + 133 5 = 144 5 .
In 1988 Noam Elkies found the first counterexample to Euler’s
conjecture for fourth powers:
2, 682, 440 4 + 15, 365, 639 4 + 18, 796, 760 4 = 20, 615, 673 4 .
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Diophantine Equations
Fermat’s Last Theorem
Euler’s Conjectures
Euler was wrong!
In 1966, L. J. Lander and T. R. Parkin found four fifth powers that
sum to a fifth power:
27 5 + 84 5 + 110 5 + 133 5 = 144 5 .
In 1988 Noam Elkies found the first counterexample to Euler’s
conjecture for fourth powers:
2, 682, 440 4 + 15, 365, 639 4 + 18, 796, 760 4 = 20, 615, 673 4 .
The smallest possible counterexample to Euler’s conjecture for
fourth powers is
95, 800 4 + 217, 519 4 + 414, 560 4 = 422, 481 4 .
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
Ekl’s Conjecture
In 1998 Randy L. Ekl revived Euler’s Conjecture by replacing the
right side of the equation with a sum of m powers:
x k 1 + · · · + x k n = y k 1 + · · · + y k m,
asserting that this equation has no solutions whenever m + n < k.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
Ekl’s Conjecture
In 1998 Randy L. Ekl revived Euler’s Conjecture by replacing the
right side of the equation with a sum of m powers:
x k 1 + · · · + x k n = y k 1 + · · · + y k m,
asserting that this equation has no solutions whenever m + n < k.
Ekl’s equation will be abbreviated (k.m.n):
◮ The first number refers to the exponent,
◮ The second to the number of terms on the right,
◮ The last number to the number of terms on the left.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
A related question
While Euler’s Conjecture asserted that the equation
x k 1 + x k 2 + · · · + x k n = y k ,
has no integer solutions when n < k, it leaves room for us to ask
the obvious question:
Must solutions exist when n = k?
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
A related question
While Euler’s Conjecture asserted that the equation
x k 1 + x k 2 + · · · + x k n = y k ,
has no integer solutions when n < k, it leaves room for us to ask
the obvious question:
Must solutions exist when n = k?
For example,
and
3 3 + 4 3 + 5 3 = 6 3
3 3 + 10 3 + 18 3 = 19 3 .
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
A related question
While Euler’s Conjecture asserted that the equation
x k 1 + x k 2 + · · · + x k n = y k ,
has no integer solutions when n < k, it leaves room for us to ask
the obvious question:
Must solutions exist when n = k?
For example,
and
3 3 + 4 3 + 5 3 = 6 3
3 3 + 10 3 + 18 3 = 19 3 .
It wasn’t until 1911 when R. Norrie assigned four biquadrates:
30 4 + 120 4 + 272 4 + 315 4 = 353 4 .
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
My work with Hillsdale senior Jonathan Gregg
Last year I worked with Jonathan
Gregg, a student at Hillsdale College,
on a problem closely related to
Euler’s Conjecture. We explored
when equations of the form
x k 1 + · · · + xk n = y k 1 + · · · + yk m
have solutions in positive whole
numbers.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
Solutions to x 3 + y 3 + z 3 = t 3
By an iterative process, we found several primitive solutions to the
equation
x 3 + y 3 + z 3 = t 3 .
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
Solutions to x 3 + y 3 + z 3 = t 3
By an iterative process, we found several primitive solutions to the
equation
x 3 + y 3 + z 3 = t 3 .
x y z t
3 4 5 6
-2 9 15 16
3 10 18 19
-2 15 33 34
1354 1389 9507 9526
22,686,834 24,248,639 162,904,689 163,229,798
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
Solutions to x 3 + y 3 + z 3 = t 3
By an iterative process, we found several primitive solutions to the
equation
x 3 + y 3 + z 3 = t 3 .
x y z t
3 4 5 6
-2 9 15 16
3 10 18 19
-2 15 33 34
1354 1389 9507 9526
22,686,834 24,248,639 162,904,689 163,229,798
The method works by solving a related equation in one variable of
degree k − 2, so it works best for small degrees k.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
Parametric Solutions to Other Equations
We found a two-parameter solution
(6a 3 + b 3 ) 3 = (6a 3 − b 3 ) 3 + 2(b 3 ) 3 + (6a 2 b) 3
and a three-parameter solution
(9a 3 + b 3 + c 3 ) 3
= (9a 3 +b 3 −c 3 ) 3 +(9a 3 −b 3 +c 3 ) 3 +(−9a 3 +b 3 +c 3 ) 3 +(6abc) 3
to the equation x 3 + y 3 + z 3 + w 3 = t 3 .
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
Parametric Solutions to Other Equations
We found a two-parameter solution
(6a 3 + b 3 ) 3 = (6a 3 − b 3 ) 3 + 2(b 3 ) 3 + (6a 2 b) 3
and a three-parameter solution
(9a 3 + b 3 + c 3 ) 3
= (9a 3 +b 3 −c 3 ) 3 +(9a 3 −b 3 +c 3 ) 3 +(−9a 3 +b 3 +c 3 ) 3 +(6abc) 3
to the equation x 3 + y 3 + z 3 + w 3 = t 3 .
We also found a four-parameter solution to the equation
x 8 1 + x 8 2 + · · · + x 8 9 = y 8 1 + y 8 2 + · · · + y 8 8,
but this slide is not large enough to contain it.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
Work still in progress
We are still trying to find any integer solution to the (6.1.6)
equation,
x 6 1 + x 6 2 + x 6 3 + x 6 4 + x 6 5 + x 6 6 = y 6 ,
for which no solutions are presently known.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
Work still in progress
We are still trying to find any integer solution to the (6.1.6)
equation,
x 6 1 + x 6 2 + x 6 3 + x 6 4 + x 6 5 + x 6 6 = y 6 ,
for which no solutions are presently known.
We are also seeking a parametric solution to the (7.1.7) equation,
x 7 1 + x 7 2 + x 7 3 + x 7 4 + x 7 5 + x 7 6 + x 7 7 = y 7 ,
for which only three solutions are presently known.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
Work still in progress
We are still trying to find any integer solution to the (6.1.6)
equation,
x 6 1 + x 6 2 + x 6 3 + x 6 4 + x 6 5 + x 6 6 = y 6 ,
for which no solutions are presently known.
We are also seeking a parametric solution to the (7.1.7) equation,
x 7 1 + x 7 2 + x 7 3 + x 7 4 + x 7 5 + x 7 6 + x 7 7 = y 7 ,
for which only three solutions are presently known.
In addition, we are working on the (5.2.2) equation,
x 5 + y 5 = z 5 + t 5 ,
which should have no solutions according to Ekl’s Conjecture.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Related Problems
Work with Jonathan Gregg
Questions Remain
References
1. Randy L. Ekl, New results in equal sums of like powers, Math.
Comp. 67 (1998), 1309–1315.
2. Noam Elkies, On A 4 + B 4 + C 4 = D 4 . Math. Comp. 51
(1988), 825–835.
3. Richard K. Guy, Unsolved Problems in Number Theory,
Springer, 1994.
4. L.J. Lander and T.R. Parkin, A counterexample to Euler’s
sum of powers conjecture. Math. Comp. 21 (1967), 446–459.
5. Jean-Charles Meyrignac, Computing Minimal Equal Sums of
Like Powers, available at http://euler.free.fr/.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Michigan NExT
◮ What? Michigan NExT is a project to support new and
recent Ph.D.s in their roles as mathematics faculty members.
In particular, the project fosters and enhances high-quality
teaching and learning of undergraduate mathematics.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Michigan NExT
◮ What? Michigan NExT is a project to support new and
recent Ph.D.s in their roles as mathematics faculty members.
In particular, the project fosters and enhances high-quality
teaching and learning of undergraduate mathematics.
◮ Who? The Michigan Section of Project NExT is open to all
full-time faculty members in mathematics departments in the
Michigan Section who have an earned doctorate.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Michigan NExT
◮ What? Michigan NExT is a project to support new and
recent Ph.D.s in their roles as mathematics faculty members.
In particular, the project fosters and enhances high-quality
teaching and learning of undergraduate mathematics.
◮ Who? The Michigan Section of Project NExT is open to all
full-time faculty members in mathematics departments in the
Michigan Section who have an earned doctorate.
◮ How? Michigan NExT is active year-long:
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Michigan NExT
◮ What? Michigan NExT is a project to support new and
recent Ph.D.s in their roles as mathematics faculty members.
In particular, the project fosters and enhances high-quality
teaching and learning of undergraduate mathematics.
◮ Who? The Michigan Section of Project NExT is open to all
full-time faculty members in mathematics departments in the
Michigan Section who have an earned doctorate.
◮ How? Michigan NExT is active year-long:
◮ Michigan NExT symposium... TODAY at 2:00 p.m.
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Michigan NExT
◮ What? Michigan NExT is a project to support new and
recent Ph.D.s in their roles as mathematics faculty members.
In particular, the project fosters and enhances high-quality
teaching and learning of undergraduate mathematics.
◮ Who? The Michigan Section of Project NExT is open to all
full-time faculty members in mathematics departments in the
Michigan Section who have an earned doctorate.
◮ How? Michigan NExT is active year-long:
◮ Michigan NExT symposium... TODAY at 2:00 p.m.
◮ Michigan Undergraduate Mathematics Conference
David Murphy
Euler’s Conjecture

Fermat and Euler
Euler’s Conjecture Revisited
Michigan NExT
Michigan NExT
◮ What? Michigan NExT is a project to support new and
recent Ph.D.s in their roles as mathematics faculty members.
In particular, the project fosters and enhances high-quality
teaching and learning of undergraduate mathematics.
◮ Who? The Michigan Section of Project NExT is open to all
full-time faculty members in mathematics departments in the
Michigan Section who have an earned doctorate.
◮ How? Michigan NExT is active year-long:
◮ Michigan NExT symposium... TODAY at 2:00 p.m.
◮ Michigan Undergraduate Mathematics Conference
◮ Michigan NExT speaker listing
David Murphy
Euler’s Conjecture