A number thoery question
posted by kate .
Please help me! Thank you very much.
Prove Fermat's Last theorem for n=3 : X^3 + Y^3 = Z^3
where X, Y, Z are rational integers, then X, Y, or Z is 0.
Hint:
* Show that if X^3 + Y^3 = Epsilon* Z^3, where X, Y, Z are quadratic integers in Q[sqrt(3)], and epsilon is a unit in Q[sqrt(3)], then X, Y, or Z is 0. (recall how to find all solutions to X^2 + Y^2 =Z^2 since it similar.)
*Begin with if X^3 + Y^3 = Epsilon* Z^3, where X, Y, Z are quadratic integers in Q[sqrt(3)], where epsilon is a unit, then lambda divides X, Y or Z, where lambda is (3+sqrt(3))/2. Also show that (lambda)^2 is an associate of 3.
*It will be useful to show that if X is congruent to 1 mod lambda, then X^3 is congruent to 1 (mod lambda)^4). Work out a similar desciption for when X is congruent to 1 mod lambda.
Use these fact to show that if X^3 + Y^3 = Epsilon* Z^3, if X and Y are not multiples of lambda, but Z is , then Z is a multiple of (lambda)^2. Do this by reducing (mod lamda)^4.
note that X^3 + Y^3 = Epsilon* Z^3 can be factored:
(x+y)(x+wy)(x+w^2*y)= Epsilon* Z^3 ,
where w is an appropriately chosen quadratic integer.
Consider each of these factors as quadratic integers p, q, r. Express x and y in terms of p, q, and r. The fact that we have three equations and two unknowns indicates there will be some extra constraint on p, q, and r.
Consider how many time lambda occurs in the prime factorization of p, q, and r. Use unique factorization of Q[sqrt(3)] to show that except for the factors of lambda that you computed , p, q, and r are cubes times units.
Use the extra constraint on p, q, and r to find another solution
X^3 + Y^3 = Epsilon* Z^3 ,
where Z has one less factor of lambda. Note that it may be necessary to exchange the roles of x, y, z, x, y, or z.
Derive a contradiction along the lines of Fermat's method of descent.
The 200 page proof overloads the buffer, sorry!
Can you show me the web link to this proof please?
Respond to this Question
Similar Questions

discrete math
1)prove that if x is rational and x not equal to 0, then 1/x is rational. 2) prove that there is a positive integers that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive? 
Math
I just wanted to check my answers with anyone willing to take the time. Identify all sets that 3/4 belongs to: a.whole #s, integers, rational #s b.rational #s c.integers, rational #s d.odd #s, whole #s, integers, rational #s Ithought … 
Math
Does 3/4 belong in the group, integers and rational numbers. in order it goes:Natural #s1,2,3,4... Whole #s0,1,2,3,4... Integerswhole #s & opposites(negatives) Rational #snumbers written in fractions or decimals Please read: (Broken … 
Precalculus
I'm HOPING this is the last problem I need help on.. :/ List the possible rational zeros of f using the rational zero theorem. f(x) = 2x^3  x^2 + 5x + 6 I applied the rational zero theorem, but none of the factors worked :( 
Math
Suppose p and q are odd integers. (a) Show that the quadratic equation x2 + px + q = 0 has no rational roots. (A number á is a “root” of that equation if: á2 + pá + q = 0. A number is rational if it is expressible as m/n for … 
Math
Identify all sets to which the number 3 belongs A. Whole numbers, integers, rational numbers B. Rational numbers C. Integers, rational numbers D. Even numbers, whole numbers, integers, rational numbers I THINK IT'S A 
Math
Proove that: a)2^(1/2) is not rational. b) prove that 2^(1/3) is not rational. I have managed to prove the first part, but i am stuck on the second can someone give me an hint please? 
Math!!! Please help!!!
Find the coefficient of x^7 for (x3)^11 Use the binomial theorem to expand (2y3x)^5 Prove that (n over r)= (n over nr) for all integers where n is greater than or equal to r and r is greater than or equal to zero Prove that (n over … 
Prealgebra
Jeremy is playing a game called “Rational Round Up” where he has to collect all the numbers in a maze that are rational and then get to the end of the maze. When he collects a number he must prove it is rational by writing it as … 
algebra
3x^4 + 5x^2  2 = 0 give imaginary and real roots rational roots theorem factors of (+)p/q are possible rational zeros of function f where the coefficients of f are integers. how do you go about solving this?