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IndustriaL
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I want to find three integers, a b and c, with a^3 + b^3 = 17 * c^3.
all each the smallest integer possible
all each the smallest integer possible
Zurtex said:Assuming your not looking for the trivial solution (0,0,0) and other trivial solutions like (1, -1, 0). Then mathematica can't find a single instance where it is true.
Actually I just used the FindInstance function for c > 0 and then for c < 0.saltydog said:Alright, I'd like to qualify Zurtx's statement if I may: Mathematica cannot find any value of a and b under 5000 which satisfy the equation. Frankly, if I had access to a faster PC I'd run it up to a million at least as well as optimize my algorithm. It kinds looks like it's related to Fermat's theorem. Is there a proof that there is no solution?
Zurtex said:Actually I just used the FindInstance function for c > 0 and then for c < 0.
matt grime said:i suspect if you have a basic knowledge of cubic number fields then the answer is known - we can do it for quadratics in quadratic number fields.
The solution to this puzzle is that there are no integer solutions. This has been proven by mathematicians using complex number theory.
One approach is to use the Pythagorean theorem in combination with the fact that the sum of two cubes can be factored into (a+b)(a^2-ab+b^2). However, this approach will not lead to any integer solutions.
No, this puzzle cannot be solved using trial and error as it involves complex numbers and requires a mathematical proof.
Yes, there are various methods and techniques used by mathematicians to approach this puzzle, such as using number theory, algebraic manipulation, and complex numbers. However, none of these methods have led to a solution for the 3 Integer Puzzle.
The 3 Integer Puzzle has significant implications in the field of mathematics as it highlights the complexity of number theory and the limitations of solving certain equations. It also emphasizes the importance of rigorous proof and the search for solutions in mathematics.