a RARE mistake from Euler (AIME 1989)  Summary and Q&A
TL;DR
The equation in the content, known as the Oilers sum of powers conjecture, was disproved in 1966 and is solved to find that the value of "n" is 144.
Questions & Answers
Q: What is the Oilers sum of powers conjecture?
The Oilers sum of powers conjecture is an equation that implies it should have no integer solutions, but it was disproved in 1966.
Q: How is the value of "n" in the equation determined?
The value of "n" is determined by making observations and performing calculations based on the given information, resulting in the conclusion that "n" is equal to 144.
Q: What is modulo arithmetic?
Modulo arithmetic is a mathematical operation that involves finding the remainder when a number is divided by another number. It can be used to determine if a number is divisible by another number.
Q: How is modulo arithmetic used in the content?
Modulo arithmetic is used in the content to determine if "n" is divisible by 2, 3, and 5, based on the remainders obtained when "n" is divided by these numbers.
Summary & Key Takeaways

The Oilers sum of powers conjecture equation, which suggests that an equation should have no integer solutions, was disproved in 1966.

The equation is solved by making observations and performing calculations to find that the value of "n" is 144.

Modulo arithmetic is demonstrated to show how it can be used to determine if a number is divisible by another number.