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Can you solve these math problems?

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Originally posted by: ActuaryTm
If you're truly seeking a challenge, given that x, y, z, and n are integers, prove:
  • x^n + y^n = z^n
Has no nonzero solutions for n>2.
Click to expand...

The proof follows a program formulated around 1985 by Frey and Serre. By classical results of Fermat, Euler, Dirichlet, Legendre, and Lame, we may assume that n=p, an odd prime >= 11. Suppose a,b,c is in Z, abc /= 0, and a^p+b^p = c^p. Without loss of generality we may assume that 2|a and b = 1 (mod 4). Frey observed that the elliptic curve E : y^2=x(x-a^p)(x+b^p) has the following "remarkable" properties:
(1) E is semistable with conductor N_E = Product(l, l|abc); and
(2) rho_E,p is unramified outside 2p and is flat at p.
By the modularity theorem of Wiles and Taylor-Wiles, there is an eigenform f included in S_2(Gamma_0(N_e)) such that rho_f,p = rho_E,p. A theorem of Mazur implies rho_E,p is irreducible, so Ribet's thereom produces a Hecke eigenform g included in S_2(Gamma_0(2)) such that rho_g,p = rho_f,p mod phi for some phi|p. But X_0(2) has genus 0, so S_2(Gamma_0(2)) = 0. This is a contradiction and Fermat's Last Theorem follows. QED.
 
Originally posted by: chuckywang
Originally posted by: ActuaryTm
If you're truly seeking a challenge, given that x, y, z, and n are integers, prove:
  • x^n + y^n = z^n
Has no nonzero solutions for n>2.
Click to expand...

The proof follows a program formulated around 1985 by Frey and Serre. By classical results of Fermat, Euler, Dirichlet, Legendre, and Lame, we may assume that n=p, an odd prime >= 11. Suppose a,b,c is in Z, abc /= 0, and a^p+b^p = c^p. Without loss of generality we may assume that 2|a and b = 1 (mod 4). Frey observed that the elliptic curve E : y^2=x(x-a^p)(x+b^p) has the following "remarkable" properties:
(1) E is semistable with conductor N_E = Product(l, l|abc); and
(2) rho_E,p is unramified outside 2p and is flat at p.
By the modularity theorem of Wiles and Taylor-Wiles, there is an eigenform f included in S_2(Gamma_0(N_e)) such that rho_f,p = rho_E,p. A theorem of Mazur implies rho_E,p is irreducible, so Ribet's thereom produces a Hecke eigenform g included in S_2(Gamma_0(2)) such that rho_g,p = rho_f,p mod phi for some phi|p. But X_0(2) has genus 0, so S_2(Gamma_0(2)) = 0. This is a contradiction and Fermat's Last Theorem follows. QED.
Click to expand...
*head explodes again*

What are you, a math major?
 
Originally posted by: ActuaryTm
If you're truly seeking a challenge, given that x, y, z, and n are integers, prove:
  • x^n + y^n = z^n
Has no nonzero solutions for n>2.
Click to expand...

I have a brilliant proof, unforunately, it cannot share it with you due to size limitations of a post.
 
Originally posted by: Leper Messiah
*head explodes again*

What are you, a math major?
Click to expand...
Nearly any student majoring in technical field requiring enough courses in mathematics has likely run across Fermat's "proof" (there is some doubt as to whether Fermat's notes in a margin constituted a proof rather than conjecture) of the Diophantine equation. A degree in mathematics isn't necessary to have exposure or knowledge of Fermat's work in this regard, but considering nearly 300 years elapsed between Fermat's statement and the first widely-recognized, rigorous proof, I doubt many (read: any) here have the mathematical prowess to generate the proof(s) of their own accord.

Not even myself (and I do have a degree in mathematics).
 
arent those basic algebra 1 and geometry problems?

i remember seing all of those in HS

if i had 10 minutes and some paper I could probably do them...i def remember the last one
 
Originally posted by: ActuaryTm
Originally posted by: Leper Messiah
*head explodes again*

What are you, a math major?
Click to expand...
Nearly any student majoring in technical field requiring enough courses in mathematics has likely run across Fermat's "proof" (there is some doubt as to whether Fermat's notes in a margin constituted a proof rather than conjecture) of the Diophantine equation. A degree in mathematics isn't necessary to have exposure or knowledge of Fermat's work in this regard, but considering nearly 300 years elapsed between Fermat's statement and the first widely-recognized, rigorous proof, I doubt many (read: any) here have the mathematical prowess to generate the proof(s) of their own accord.

Not even myself (and I do have a degree in mathematics).
Click to expand...

The number of people that understand the proof of Fermat's Last Theorem can be counted on the fingers of your hands.
 
1) n^2-19n+99=q^2 ==> n^2-19n+99-q^2 = 0. Solve this using the quadratic formula, and you get the discriminent as: 4q^2-35. This would have to be an perfect square since we are looking for integral solutions. Therefore 4q^2-35 = m^2, for some integer m. This simplifies to: (2q-m)*(2q+m)=35. Therefore, the only solutions are:

a) 2q+m=35, 2q-m = 1, so q=9
b) 2q+m=7, 2q-m=5, so q=3.

For the q=9 case, we get n=1, 18. For the q=3 case, we get n=9, 10. Therefore the sum of all possible values of n such that n^2-19n+99 is a perfect square is: 1+18+9+10 = 38.

 
2) The answer is 103. Man, I had to go through a lot of work to get that, so I don't think my way was the best way. What's the shortcut in this problem, Vertimus?
 
Originally posted by: Vertimus
Continuation of the old thread.

These are selected problems I've done in the past few days that I've found interesting. Let's see how the ATOT crowd handles it!

1. Find the sum of all positive integers n for which n^2-19n+99 is a perfect square.

2. In trapezoid ABCD, leg BC is perpendicular to bases AB and CD, and diagonals AC and BD are perpendicular. Given that AB=sqrt(11) and AD=sqrt(1001), find BC^2.

3. The points A, B, and C lie on the surface of a sphere with center O and radius 20. It is given that AB=13, BC=14, CA=15, and that the distance from O to triangle ABC is x. Find x in exact terms.

4. The function f is defined by f(x)=(ax+b)/(cx+d), where a, b, c, and d are nonzero real numbers, has the properties f(19)=19, f(97)=97, and f(f(x))=x for all values of x except -d/c. Find the unique number that is not in the range of f.

5. Consider the polynomials P(x)=x^6-x^5-x^3-x^2-x and Q(x)=x^4-x^3-x^2-1. Given that p, q, r, and s are the roots of Q(x)=0, find P(p)+P(q)+P(r)+P(s).

6 A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects on eof the two verticies where it is not currently located, and crawls along a side of the tirangle to that vetex. Given that the probability that the bug moves to its starting vertex on its tenth move is x, find x.

7. Given that z is a complex number such that z+1/z=2 cos(3), find the least integer that is greater than z^2000+1/z^2000. (Angles are in degrees in this problem)

8. Let v and w be distinct, randomly chosen roots of the equation z^1997-1=0. Let x be the probability that sqrt(2+sqrt(3)) is lesser or equal to the absolute value of (v+w). Find x.

NOTE: Please include solutions with your answers. Tell me how you arrived at your answer.

I solved all of these problems without calculators or computers so you should too. They become too easy and not fun when you use calculators, and are designed without them.

Good Luck! :beer:
Click to expand...

1. 5
2. 0

3. 0
4. U
5. 7
6. S
7. 1
8. D
9. 3
 
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