Jawai Theorem
Equivalent statements of the theorem[edit]
There are several alternative ways to state Fermat's Last Theorem that are mathematically equivalent to the original statement of the problem.
In order to state them, we use mathematical notation: let N be the set of natural numbers 1, 2, 3, ..., let Z be the set of integers 0, ±1, ±2, ..., and let Q be the set of rational numbers a/b where a and b are in Z with b≠0. In what follows we will call a solution to xn + yn = zn where one or more of x, y, or z is zero a trivial solution. A solution where all three are non-zero will be called a non-trivial solution.
For comparison's sake we start with the original formulation.
- Original statement. With n, x, y, z ∈ N (meaning n, x, y, z are all positive whole numbers) and n > 2 the equation xn + yn = zn has no solutions.
Most popular treatments of the subject state it this way. In contrast, almost all math textbooks state it over Z:[citation needed]
- Equivalent statement 1: xn + yn = zn, where integer n ≥ 3, has no non-trivial solutions x, y, z ∈ Z.
The equivalence is clear if n is even. If n is odd and all three of x, y, z are negative then we can replace x, y, z with −x, −y, −z to obtain a solution in N. If two of them are negative, it must be x and z or y and z. If x, z are negative and y is positive, then we can rearrange to get (−z)n + yn = (−x)n resulting in a solution in N; the other case is dealt with analogously. Now if just one is negative, it must be x or y. If x is negative, and y and z are positive, then it can be rearranged to get (−x)n + zn = yn again resulting in a solution in N; if y is negative, the result follows symmetrically. Thus in all cases a nontrivial solution in Z would also mean a solution exists in N, the original formulation of the problem.
- Equivalent statement 2: xn + yn = zn, where integer n ≥ 3, has no non-trivial solutions x, y, z ∈ Q.
This is because the exponent of x, y and z are equal (to n), so if there is a solution in Q then it can be multiplied through by an appropriate common denominator to get a solution in Z, and hence in N.
- Equivalent statement 3: xn + yn = 1, where integer n ≥ 3, has no non-trivial solutions x, y ∈ Q.
A non-trivial solution a, b, c ∈ Z to xn + yn = zn yields the non-trivial solution a/c, b/c ∈ Q for vn + wn = 1. Conversely, a solution a/b, c/d ∈ Q to vn + wn = 1 yields the non-trivial solution ad, cb, bd for xn + yn = zn.
This last formulation is particularly fruitful, because it reduces the problem from a problem about surfaces in three dimensions to a problem about curves in two dimensions. Furthermore, it allows working over the field Q, rather than over the ring Z; fields exhibit more structure than rings, which allows for deeper analysis of their elements.
- Equivalent statement 4 – connection to elliptic curves: If a, b, c is a non-trivial solution to xp + yp = zp , p odd prime, then y2 = x(x − ap)(x + bp) (Frey curve) will be an elliptic curve.
Examining this elliptic curve with Ribet's theorem shows that it does not have a modular form. However, the proof by Andrew Wiles proves that any equation of the form y2 = x(x − an)(x + bn) does have a modular form. Any non-trivial solution to xp + yp = zp (with p an odd prime) would therefore create a contradiction, which in turn proves that no non-trivial solutions exist.
In other words, any solution that could contradict Fermat's Last Theorem could also be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true, then it would follow that no contradiction to Fermat's Last Theorem could exist either. As described above, the discovery of this equivalent statement was crucial to the eventual solution of Fermat's Last Theorem, as it provided a means by which it could be 'attacked' for all numbers at once.
This theorem is proposed by Prof Dr. Jawai, It is proposed in year of 2016 this theorem discovered in his B.Tech Career
References
Jawai Theorem
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