Fermat's Theorem: A Math Problem and the Smart Boy Who Proved It Wrong

from: http://www.math.utah.edu/~cherk/puzzles.html

Fermat, computers, and a smart boy
A computer scientist claims that he proved somehow that the Fermat theorem is correct for the following 3 numbers:

x=2233445566,
y=7788990011,
z=9988776655

He announces these 3 numbers and calls for a press conference where he is going to present the value of N (to show that

x^N + y^N = z^N

and that the guy from Princeton was wrong). As the press conference starts, a 10-years old boy raises his hand and says that the respectable scientist has made a mistake and the Fermat theorem cannot hold for those 3 numbers. The scientist checks his computer calculations and finds a bug.

How did the boy figure out that the scientist was wrong?

I am stumped, I noticed the pattern in the digits of the numbers, but I do not see how I can link that to the possibility of forming such a statement with those numbers when n is greater than 2.

x==1Mod 5, Y==1 Mod 5, Z==0 Mod 5.

It is impressive that a 10-year-old boy was able to spot the mistake in the computer scientist's proof of Fermat's Theorem. This is a testament to the power of critical thinking and intuition in mathematics. While the specific details of how the boy figured out the mistake are not mentioned, there are a few possible explanations.

First, it is possible that the boy had a deep understanding of the patterns and properties of numbers, which allowed him to recognize that the numbers chosen by the scientist did not follow the necessary conditions for Fermat's Theorem to hold. For example, Fermat's Theorem only holds for positive integers, and it is possible that the boy noticed that one or more of the numbers chosen were not positive integers.

Another possibility is that the boy had prior knowledge of Fermat's Theorem and recognized that the numbers chosen did not fit the equation. Perhaps he had studied the theorem in school or had come across it in his own mathematical explorations. This prior knowledge and understanding allowed him to quickly spot the mistake in the scientist's proof.

Finally, it is also possible that the boy simply had a keen eye for detail and noticed a discrepancy in the numbers presented. He may have noticed that the numbers did not follow a certain pattern or did not seem to fit the equation given. This shows the importance of paying attention to every detail in mathematics, as even the smallest mistake can lead to a faulty proof.

In any case, the boy's ability to spot the mistake in the scientist's proof is impressive and serves as a reminder that anyone, regardless of age or background, can contribute to the world of mathematics. It also highlights the importance of double-checking and verifying proofs, as even the most respected scientists and mathematicians can make mistakes.