Euler's Sum of Like Powers Conjecture history and related problems
Euler's Sum of Like Powers Conjecture history and related problems
Euler's Sum of Like Powers Conjecture history and related problems
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Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Euler’s <strong>Sum</strong> <strong>of</strong> <strong>Like</strong> <strong>Powers</strong> <strong>Conjecture</strong><br />
<strong>history</strong> <strong>and</strong> <strong>related</strong> <strong>problems</strong><br />
2011 Michigan MAA Section Meeting<br />
Western Michigan University<br />
David Murphy<br />
Hillsdale College<br />
May 7, 2011<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Diophantus <strong>and</strong> Diophantine equations<br />
Little is known about Diophantus <strong>of</strong> Alex<strong>and</strong>ria, sometimes called<br />
the “father <strong>of</strong> algebra.”<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Diophantus <strong>and</strong> Diophantine equations<br />
Little is known about Diophantus <strong>of</strong> Alex<strong>and</strong>ria, sometimes called<br />
the “father <strong>of</strong> algebra.”<br />
◮ He lived between 150 B.C. <strong>and</strong> 350 A.D.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Diophantus <strong>and</strong> Diophantine equations<br />
Little is known about Diophantus <strong>of</strong> Alex<strong>and</strong>ria, sometimes called<br />
the “father <strong>of</strong> algebra.”<br />
◮ He lived between 150 B.C. <strong>and</strong> 350 A.D.<br />
◮ “...his boyhood lasted 1/6th <strong>of</strong> his life; he married after 1/7th<br />
more; his beard grew after 1/12th more, <strong>and</strong> his son was born<br />
5 years later; the son lived to half his father’s age, <strong>and</strong> the<br />
father died 4 years after the son” – 500 A.D.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Diophantus <strong>and</strong> Diophantine equations<br />
Little is known about Diophantus <strong>of</strong> Alex<strong>and</strong>ria, sometimes called<br />
the “father <strong>of</strong> algebra.”<br />
◮ He lived between 150 B.C. <strong>and</strong> 350 A.D.<br />
◮ “...his boyhood lasted 1/6th <strong>of</strong> his life; he married after 1/7th<br />
more; his beard grew after 1/12th more, <strong>and</strong> his son was born<br />
5 years later; the son lived to half his father’s age, <strong>and</strong> the<br />
father died 4 years after the son” – 500 A.D.<br />
◮ He wrote Arithmetica, a collection <strong>of</strong> 189 <strong>problems</strong> in number<br />
theory that require positive rational solutions.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Diophantus <strong>and</strong> Diophantine equations<br />
Little is known about Diophantus <strong>of</strong> Alex<strong>and</strong>ria, sometimes called<br />
the “father <strong>of</strong> algebra.”<br />
◮ He lived between 150 B.C. <strong>and</strong> 350 A.D.<br />
◮ “...his boyhood lasted 1/6th <strong>of</strong> his life; he married after 1/7th<br />
more; his beard grew after 1/12th more, <strong>and</strong> his son was born<br />
5 years later; the son lived to half his father’s age, <strong>and</strong> the<br />
father died 4 years after the son” – 500 A.D.<br />
◮ He wrote Arithmetica, a collection <strong>of</strong> 189 <strong>problems</strong> in number<br />
theory that require positive rational solutions.<br />
From Book II, Problem 8 <strong>of</strong> his Arithmetica:<br />
To divide a given square into two squares.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Pierre de Fermat (1601–1665)<br />
“It is impossible for a cube to be the sum <strong>of</strong> two cubes,<br />
a fourth power to be the sum <strong>of</strong> two fourth powers, or<br />
in general for any number that is a power greater than<br />
the second to be the sum <strong>of</strong> two like powers. I have<br />
discovered a truly marvelous demonstration <strong>of</strong> this<br />
proposition that this margin is too narrow to contain.”<br />
— Fermat, 1630’s<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Pierre de Fermat (1601–1665)<br />
“It is impossible for a cube to be the sum <strong>of</strong> two cubes,<br />
a fourth power to be the sum <strong>of</strong> two fourth powers, or<br />
in general for any number that is a power greater than<br />
the second to be the sum <strong>of</strong> two like powers. I have<br />
discovered a truly marvelous demonstration <strong>of</strong> this<br />
proposition that this margin is too narrow to contain.”<br />
— Fermat, 1630’s<br />
Fermat’s Last Theorem<br />
The equation<br />
x n + y n = z n<br />
has no solutions in positive whole numbers whenever n ≥ 3.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
The end <strong>of</strong> a 350-year-old problem<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Leonhard Euler (1707–1783)<br />
“It has seemed to many Geometers that this<br />
theorem [Fermat’s Last ‘Theorem’] may be<br />
generalized. Just as there do not exist two<br />
cubes whose sum or difference is a cube,<br />
it is certain that it is impossible to exhibit three<br />
biquadrates whose sum is a biquadrate, but that at least four<br />
biquadrates are needed if their sum is to be a biquadrate, although<br />
no one has been able up to the present to assign four such<br />
biquadrates. In the same manner it would seem impossible to<br />
exhibit four fifth powers whose sum is a fifth power, <strong>and</strong> similarly<br />
for higher powers.” — Euler, 1769<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Euler’s <strong>Conjecture</strong><br />
The equation<br />
x k 1 + x k 2 + · · · + x k n = y k .<br />
has no solutions in positive whole numbers whenever n < k.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Euler’s <strong>Conjecture</strong><br />
The equation<br />
x k 1 + x k 2 + · · · + x k n = y k .<br />
has no solutions in positive whole numbers whenever n < k.<br />
Notice that if Euler’s <strong>Conjecture</strong> is true, then Fermat’s Last<br />
Theorem is the special case where n = 2 <strong>and</strong> k > 2.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Euler’s <strong>Conjecture</strong><br />
The equation<br />
x k 1 + x k 2 + · · · + x k n = y k .<br />
has no solutions in positive whole numbers whenever n < k.<br />
Notice that if Euler’s <strong>Conjecture</strong> is true, then Fermat’s Last<br />
Theorem is the special case where n = 2 <strong>and</strong> k > 2.<br />
For nearly 200 years, no significant progress was made on Euler’s<br />
<strong>Conjecture</strong>.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Euler was wrong!<br />
In 1966, L. J. L<strong>and</strong>er <strong>and</strong> T. R. Parkin found four fifth powers that<br />
sum to a fifth power:<br />
27 5 + 84 5 + 110 5 + 133 5 = 144 5 .<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Euler was wrong!<br />
In 1966, L. J. L<strong>and</strong>er <strong>and</strong> T. R. Parkin found four fifth powers that<br />
sum to a fifth power:<br />
27 5 + 84 5 + 110 5 + 133 5 = 144 5 .<br />
In 1988 Noam Elkies found the first counterexample to Euler’s<br />
conjecture for fourth powers:<br />
2, 682, 440 4 + 15, 365, 639 4 + 18, 796, 760 4 = 20, 615, 673 4 .<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Diophantine Equations<br />
Fermat’s Last Theorem<br />
Euler’s <strong>Conjecture</strong>s<br />
Euler was wrong!<br />
In 1966, L. J. L<strong>and</strong>er <strong>and</strong> T. R. Parkin found four fifth powers that<br />
sum to a fifth power:<br />
27 5 + 84 5 + 110 5 + 133 5 = 144 5 .<br />
In 1988 Noam Elkies found the first counterexample to Euler’s<br />
conjecture for fourth powers:<br />
2, 682, 440 4 + 15, 365, 639 4 + 18, 796, 760 4 = 20, 615, 673 4 .<br />
The smallest possible counterexample to Euler’s conjecture for<br />
fourth powers is<br />
95, 800 4 + 217, 519 4 + 414, 560 4 = 422, 481 4 .<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
Ekl’s <strong>Conjecture</strong><br />
In 1998 R<strong>and</strong>y L. Ekl revived Euler’s <strong>Conjecture</strong> by replacing the<br />
right side <strong>of</strong> the equation with a sum <strong>of</strong> m powers:<br />
x k 1 + · · · + x k n = y k 1 + · · · + y k m,<br />
asserting that this equation has no solutions whenever m + n < k.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
Ekl’s <strong>Conjecture</strong><br />
In 1998 R<strong>and</strong>y L. Ekl revived Euler’s <strong>Conjecture</strong> by replacing the<br />
right side <strong>of</strong> the equation with a sum <strong>of</strong> m powers:<br />
x k 1 + · · · + x k n = y k 1 + · · · + y k m,<br />
asserting that this equation has no solutions whenever m + n < k.<br />
Ekl’s equation will be abbreviated (k.m.n):<br />
◮ The first number refers to the exponent,<br />
◮ The second to the number <strong>of</strong> terms on the right,<br />
◮ The last number to the number <strong>of</strong> terms on the left.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
A <strong>related</strong> question<br />
While Euler’s <strong>Conjecture</strong> asserted that the equation<br />
x k 1 + x k 2 + · · · + x k n = y k ,<br />
has no integer solutions when n < k, it leaves room for us to ask<br />
the obvious question:<br />
Must solutions exist when n = k?<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
A <strong>related</strong> question<br />
While Euler’s <strong>Conjecture</strong> asserted that the equation<br />
x k 1 + x k 2 + · · · + x k n = y k ,<br />
has no integer solutions when n < k, it leaves room for us to ask<br />
the obvious question:<br />
Must solutions exist when n = k?<br />
For example,<br />
<strong>and</strong><br />
3 3 + 4 3 + 5 3 = 6 3<br />
3 3 + 10 3 + 18 3 = 19 3 .<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
A <strong>related</strong> question<br />
While Euler’s <strong>Conjecture</strong> asserted that the equation<br />
x k 1 + x k 2 + · · · + x k n = y k ,<br />
has no integer solutions when n < k, it leaves room for us to ask<br />
the obvious question:<br />
Must solutions exist when n = k?<br />
For example,<br />
<strong>and</strong><br />
3 3 + 4 3 + 5 3 = 6 3<br />
3 3 + 10 3 + 18 3 = 19 3 .<br />
It wasn’t until 1911 when R. Norrie assigned four biquadrates:<br />
30 4 + 120 4 + 272 4 + 315 4 = 353 4 .<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
My work with Hillsdale senior Jonathan Gregg<br />
Last year I worked with Jonathan<br />
Gregg, a student at Hillsdale College,<br />
on a problem closely <strong>related</strong> to<br />
Euler’s <strong>Conjecture</strong>. We explored<br />
when equations <strong>of</strong> the form<br />
x k 1 + · · · + xk n = y k 1 + · · · + yk m<br />
have solutions in positive whole<br />
numbers.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
Solutions to x 3 + y 3 + z 3 = t 3<br />
By an iterative process, we found several primitive solutions to the<br />
equation<br />
x 3 + y 3 + z 3 = t 3 .<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
Solutions to x 3 + y 3 + z 3 = t 3<br />
By an iterative process, we found several primitive solutions to the<br />
equation<br />
x 3 + y 3 + z 3 = t 3 .<br />
x y z t<br />
3 4 5 6<br />
-2 9 15 16<br />
3 10 18 19<br />
-2 15 33 34<br />
1354 1389 9507 9526<br />
22,686,834 24,248,639 162,904,689 163,229,798<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
Solutions to x 3 + y 3 + z 3 = t 3<br />
By an iterative process, we found several primitive solutions to the<br />
equation<br />
x 3 + y 3 + z 3 = t 3 .<br />
x y z t<br />
3 4 5 6<br />
-2 9 15 16<br />
3 10 18 19<br />
-2 15 33 34<br />
1354 1389 9507 9526<br />
22,686,834 24,248,639 162,904,689 163,229,798<br />
The method works by solving a <strong>related</strong> equation in one variable <strong>of</strong><br />
degree k − 2, so it works best for small degrees k.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
Parametric Solutions to Other Equations<br />
We found a two-parameter solution<br />
(6a 3 + b 3 ) 3 = (6a 3 − b 3 ) 3 + 2(b 3 ) 3 + (6a 2 b) 3<br />
<strong>and</strong> a three-parameter solution<br />
(9a 3 + b 3 + c 3 ) 3<br />
= (9a 3 +b 3 −c 3 ) 3 +(9a 3 −b 3 +c 3 ) 3 +(−9a 3 +b 3 +c 3 ) 3 +(6abc) 3<br />
to the equation x 3 + y 3 + z 3 + w 3 = t 3 .<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
Parametric Solutions to Other Equations<br />
We found a two-parameter solution<br />
(6a 3 + b 3 ) 3 = (6a 3 − b 3 ) 3 + 2(b 3 ) 3 + (6a 2 b) 3<br />
<strong>and</strong> a three-parameter solution<br />
(9a 3 + b 3 + c 3 ) 3<br />
= (9a 3 +b 3 −c 3 ) 3 +(9a 3 −b 3 +c 3 ) 3 +(−9a 3 +b 3 +c 3 ) 3 +(6abc) 3<br />
to the equation x 3 + y 3 + z 3 + w 3 = t 3 .<br />
We also found a four-parameter solution to the equation<br />
x 8 1 + x 8 2 + · · · + x 8 9 = y 8 1 + y 8 2 + · · · + y 8 8,<br />
but this slide is not large enough to contain it.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
Work still in progress<br />
We are still trying to find any integer solution to the (6.1.6)<br />
equation,<br />
x 6 1 + x 6 2 + x 6 3 + x 6 4 + x 6 5 + x 6 6 = y 6 ,<br />
for which no solutions are presently known.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
Work still in progress<br />
We are still trying to find any integer solution to the (6.1.6)<br />
equation,<br />
x 6 1 + x 6 2 + x 6 3 + x 6 4 + x 6 5 + x 6 6 = y 6 ,<br />
for which no solutions are presently known.<br />
We are also seeking a parametric solution to the (7.1.7) equation,<br />
x 7 1 + x 7 2 + x 7 3 + x 7 4 + x 7 5 + x 7 6 + x 7 7 = y 7 ,<br />
for which only three solutions are presently known.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
Work still in progress<br />
We are still trying to find any integer solution to the (6.1.6)<br />
equation,<br />
x 6 1 + x 6 2 + x 6 3 + x 6 4 + x 6 5 + x 6 6 = y 6 ,<br />
for which no solutions are presently known.<br />
We are also seeking a parametric solution to the (7.1.7) equation,<br />
x 7 1 + x 7 2 + x 7 3 + x 7 4 + x 7 5 + x 7 6 + x 7 7 = y 7 ,<br />
for which only three solutions are presently known.<br />
In addition, we are working on the (5.2.2) equation,<br />
x 5 + y 5 = z 5 + t 5 ,<br />
which should have no solutions according to Ekl’s <strong>Conjecture</strong>.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Related Problems<br />
Work with Jonathan Gregg<br />
Questions Remain<br />
References<br />
1. R<strong>and</strong>y L. Ekl, New results in equal sums <strong>of</strong> like powers, Math.<br />
Comp. 67 (1998), 1309–1315.<br />
2. Noam Elkies, On A 4 + B 4 + C 4 = D 4 . Math. Comp. 51<br />
(1988), 825–835.<br />
3. Richard K. Guy, Unsolved Problems in Number Theory,<br />
Springer, 1994.<br />
4. L.J. L<strong>and</strong>er <strong>and</strong> T.R. Parkin, A counterexample to Euler’s<br />
sum <strong>of</strong> powers conjecture. Math. Comp. 21 (1967), 446–459.<br />
5. Jean-Charles Meyrignac, Computing Minimal Equal <strong>Sum</strong>s <strong>of</strong><br />
<strong>Like</strong> <strong>Powers</strong>, available at http://euler.free.fr/.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Michigan NExT<br />
◮ What? Michigan NExT is a project to support new <strong>and</strong><br />
recent Ph.D.s in their roles as mathematics faculty members.<br />
In particular, the project fosters <strong>and</strong> enhances high-quality<br />
teaching <strong>and</strong> learning <strong>of</strong> undergraduate mathematics.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Michigan NExT<br />
◮ What? Michigan NExT is a project to support new <strong>and</strong><br />
recent Ph.D.s in their roles as mathematics faculty members.<br />
In particular, the project fosters <strong>and</strong> enhances high-quality<br />
teaching <strong>and</strong> learning <strong>of</strong> undergraduate mathematics.<br />
◮ Who? The Michigan Section <strong>of</strong> Project NExT is open to all<br />
full-time faculty members in mathematics departments in the<br />
Michigan Section who have an earned doctorate.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Michigan NExT<br />
◮ What? Michigan NExT is a project to support new <strong>and</strong><br />
recent Ph.D.s in their roles as mathematics faculty members.<br />
In particular, the project fosters <strong>and</strong> enhances high-quality<br />
teaching <strong>and</strong> learning <strong>of</strong> undergraduate mathematics.<br />
◮ Who? The Michigan Section <strong>of</strong> Project NExT is open to all<br />
full-time faculty members in mathematics departments in the<br />
Michigan Section who have an earned doctorate.<br />
◮ How? Michigan NExT is active year-long:<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Michigan NExT<br />
◮ What? Michigan NExT is a project to support new <strong>and</strong><br />
recent Ph.D.s in their roles as mathematics faculty members.<br />
In particular, the project fosters <strong>and</strong> enhances high-quality<br />
teaching <strong>and</strong> learning <strong>of</strong> undergraduate mathematics.<br />
◮ Who? The Michigan Section <strong>of</strong> Project NExT is open to all<br />
full-time faculty members in mathematics departments in the<br />
Michigan Section who have an earned doctorate.<br />
◮ How? Michigan NExT is active year-long:<br />
◮ Michigan NExT symposium... TODAY at 2:00 p.m.<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Michigan NExT<br />
◮ What? Michigan NExT is a project to support new <strong>and</strong><br />
recent Ph.D.s in their roles as mathematics faculty members.<br />
In particular, the project fosters <strong>and</strong> enhances high-quality<br />
teaching <strong>and</strong> learning <strong>of</strong> undergraduate mathematics.<br />
◮ Who? The Michigan Section <strong>of</strong> Project NExT is open to all<br />
full-time faculty members in mathematics departments in the<br />
Michigan Section who have an earned doctorate.<br />
◮ How? Michigan NExT is active year-long:<br />
◮ Michigan NExT symposium... TODAY at 2:00 p.m.<br />
◮ Michigan Undergraduate Mathematics Conference<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>
Fermat <strong>and</strong> Euler<br />
Euler’s <strong>Conjecture</strong> Revisited<br />
Michigan NExT<br />
Michigan NExT<br />
◮ What? Michigan NExT is a project to support new <strong>and</strong><br />
recent Ph.D.s in their roles as mathematics faculty members.<br />
In particular, the project fosters <strong>and</strong> enhances high-quality<br />
teaching <strong>and</strong> learning <strong>of</strong> undergraduate mathematics.<br />
◮ Who? The Michigan Section <strong>of</strong> Project NExT is open to all<br />
full-time faculty members in mathematics departments in the<br />
Michigan Section who have an earned doctorate.<br />
◮ How? Michigan NExT is active year-long:<br />
◮ Michigan NExT symposium... TODAY at 2:00 p.m.<br />
◮ Michigan Undergraduate Mathematics Conference<br />
◮ Michigan NExT speaker listing<br />
David Murphy<br />
Euler’s <strong>Conjecture</strong>