In this article we are going to test the mathematical knowledge of ChatGPT. We will try to take advantage of artificial intelligence to find a counterexample of the Fundamental Theorem of Algebra, a discovery that would undoubtedly launch us towards the Fields Medal.
If we ask for the roots of a polynomial of degree 3, in this case all real, ChatGPT argues that the analytical resolution can be complicated depending on the proposed polynomial, so it recommends us to use a iterative numerical method like the Newton-Raphson method.
An error in the calculation of the derivative
So far, we cannot doubt the mathematical ability of the AI, so we try to solve the problem of finding the roots of the polynomial p(x) = x3 – 3x2 + 4 and to our surprise it performs the calculation of the derivative incorrectly, so obtaining the roots is not correct. It returns x = 0 as the root of the polynomial and we ask it to check it. Of course, he realizes that there is a mistake, but he doesn’t know where he made it. We help by showing that the error is in the derivative of the polynomial and we ask him to recalculate the roots using the Newton–Raphson method. Surprisingly, he miscalculates again. This time in a simple operation as we can see in the following image:
Noticing the error in the calculations, we ask him again making another mistake, so we give him the first iteration of the Newton-Raphson method, namely x₁ = 5/3 and we ask to continue the iterations, resulting in x₁ = 5/3 is the root of the polynomial. We corroborate by asking again if the value 5/3 is a root of the polynomial, and we obtain an affirmative answer. We ask to calculate the value of the polynomial at that value, and, since the result is different from zero, we show it that it cannot be a root. He understands and apologizes as we can see below:
We conclude that the theory about the Newton-Raphson method is correct, but its application is not, so we try to find the roots using another method, such as the factorization of the polynomial.
In this case, it returns us that the roots of the polynomial p(x) are x = r and x = 1 ± 2i.
When asked to verify that the value of p(1+2i) is different from zero and therefore cannot be a root of our polynomial, it again acknowledges the error. Arriving at this situation, we go with a clue, and we tell him that x = – 1 is a real root of the polynomial and that he calculates the rest of the roots. His first answer could not be more surprising, telling us that in addition to x = – 1, the other roots of the polynomial p (x) = 4 – 3x2 + x3 are x = 1 + 2i and x = 1 – 2i. Up to four times it gives incorrect results again, so we have no choice but to provide it with a new root. In this case, instead of giving it, we ask if x = 2 is the root of our polynomial. Judge for yourselves the answer, or rather, the calculations that ChatGPT performs to check that x = 2 is not a root:
After checking your calculations again, finish by explaining that the roots of our polynomial are x = – 1, x = 1, and x = 2.
We ask it to verify that all three are roots, returning the correct one, that is, x = 1 is not a root while the other two values are. We do not give up and try to find the third root of the polynomial, and it is precisely where we find the biggest misinterpretation. He explains that: “The Fundamental Theorem of Algebra establishes that every polynomial of degree n has exactly n roots, both real and complex.
Thus, if a polynomial of degree n has k real roots, then it must have nk complex roots. So far we agree. We continue with: “In the case of the polynomial p(x) = 4 – 3x2 + x3,we have found that it has two real roots, x = – 1 and x = 2. Since p(x) is a polynomial of degree 3, then it must have an additional complex root. This complex root is x = 1 ± 2i.” We cannot get out of our astonishment and we understand that he only wanted to show us one of the two roots, so we give him another chance, resulting in:
So if we’re right, we’ve just found a counterexample to Fundamental Theorem of Algebra, a polynomial of degree 3 with 4 roots. Do we propose for the fields medal?
Artificial intelligence reaffirms that its answer is correct up to two more times, showing that a degree 3 polynomial can have 4 roots. He even recommends us to find them through the Bisection method. Now yes, we give up looking for the roots of a simple degree 3 polynomial. We cordially say goodbye with one last pill:
As a final summary, we are not saying that ChatGPT is a bad artificial intelligenceFar from it, if not just the opposite, it is very good. But in his own, in Natural Language Processing, although in Mathematics, he still has a lot to learn. We must be critical of the results that the engines return to us: they are not true no matter how well explained they are, and a human is always needed to verify their veracity.
