HapaxOromenon2
Badges: 2
Rep:
?
#1
Report Thread starter 3 years ago
#1
Prove that the equation x^n + y^n = z^n, where n is an integer > 1, has no solution in integers x, y, and z, with 0 < x <= n, 0 < y <= n.

Find the shortest solution you can; I have one that only takes 4 lines of working.
0
reply
joostan
Badges: 13
Rep:
?
#2
Report 3 years ago
#2
(Original post by HapaxOromenon2)
Prove that the equation x^n + y^n = z^n, where n is an integer > 1, has no solution in integers x, y, and z, with 0 < x <= n, 0 < y <= n.

Find the shortest solution you can; I have one that only takes 4 lines of working.
One can get this pretty snappily with the contrapositive. Took me quite a while to notice this.
Spoiler:
Show
Suppose the equation has a solution, and w.l.o.g. x \leq y &lt; z.
x^n=z^n-y^n=(z-y)\displaystyle\sum_{k=0}^{n-1}z^{n-1-k}y^k &gt; (z-y)ny^{n-1} \geq nx^{n-1}.
But x^n &gt; nx^{n-1} \iff x&gt;n. The result follows.
0
reply
HapaxOromenon2
Badges: 2
Rep:
?
#3
Report Thread starter 3 years ago
#3
(Original post by joostan)
One can get this pretty snappily with the contrapositive. Took me quite a while to notice this.
Spoiler:
Show
Suppose the equation has a solution, and w.l.o.g. x \leq y &lt; z.
x^n=z^n-y^n=(z-y)\displaystyle\sum_{k=0}^{n-1}z^{n-1-k}y^k &gt; (z-y)ny^{n-1} \geq nx^{n-1}.
But x^n &gt; nx^{n-1} \iff x&gt;n. The result follows.
What I did: By Fermat's Last Theorem there are no solutions for n>2, so since we are given n>1, we need only consider n = 2. Thus we have that x^2 + y^2 = z^2 and 0<x<=2, 0<y<=2, so x and y can each only be 1 or 2.
Thus we check 1^2 + 1^2 = 2, 1^2 + 2^2 = 2^2 + 1^2 = 5, 2^2 + 2^2 = 8, none of which are square numbers. Thus the result follows.

Much simpler than your approach...
0
reply
joostan
Badges: 13
Rep:
?
#4
Report 3 years ago
#4
(Original post by HapaxOromenon2)
What I did: By Fermat's Last Theorem there are no solutions for n>2, so since we are given n>1, we need only consider n = 2. Thus we have that x^2 + y^2 = z^2 and 0<x<=2, 0<y<=2, so x and y can each only be 1 or 2.
Thus we check 1^2 + 1^2 = 2, 1^2 + 2^2 = 2^2 + 1^2 = 5, 2^2 + 2^2 = 8, none of which are square numbers. Thus the result follows.

Much simpler than your approach...
I disagree, assuming Fermat's Last Theorem is gigantic, the proof of it - monumental.
Moreover this question was initially set before the theorem was proved.
0
reply
HapaxOromenon2
Badges: 2
Rep:
?
#5
Report Thread starter 3 years ago
#5
(Original post by joostan)
I disagree, assuming Fermat's Last Theorem is gigantic, the proof of it - monumental.
Moreover this question was initially set before the theorem was proved.
True. But it's still easier to understand than the thing you did, because you don't have to understand the proof of a theorem in order to understand it's statement and to use it to solve problems.
0
reply
Zacken
Badges: 22
Rep:
?
#6
Report 3 years ago
#6
(Original post by joostan)
I disagree, assuming Fermat's Last Theorem is gigantic, the proof of it - monumental.
Moreover this question was initially set before the theorem was proved.
He's a well known troll, this is his ~10/11th account. I wouldn't bother replying to him.
0
reply
HapaxOromenon2
Badges: 2
Rep:
?
#7
Report Thread starter 3 years ago
#7
(Original post by Zacken)
He's a well known troll, this is his ~10/11th account. I wouldn't bother replying to him.
I demand that you cease and desist from defamation of me at once.
0
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Back
to top
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

University open days

  • University of Suffolk
    Undergraduate Open Day - Ipswich Main Campus Undergraduate
    Mon, 9 Dec '19
  • University of Hertfordshire
    All Subjects Undergraduate
    Wed, 11 Dec '19
  • University of Lincoln
    Undergraduate Open Day Undergraduate
    Wed, 11 Dec '19

Which party will you be voting for in the General Election?

Conservatives (124)
20.13%
Labour (308)
50%
Liberal Democrats (83)
13.47%
Green Party (30)
4.87%
Brexit Party (7)
1.14%
Independent Group for Change (Change UK) (3)
0.49%
SNP (13)
2.11%
Plaid Cymru (3)
0.49%
Democratic Unionist Party (DUP) (0)
0%
Sinn Fein (6)
0.97%
SDLP (0)
0%
Ulster Unionist (3)
0.49%
UKIP (8)
1.3%
Other (4)
0.65%
None (24)
3.9%

Watched Threads

View All