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Shades of Fermat's Last Theorem

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

Exhaustion

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

Discrete Trends

Age 16 to 18 Challenge Level:

Why do this problem?
It gives practice in working with inequalities.

As we know $n$ is a positive integer learners can investigate $n^{{1\over n}}$ for for different values of $n$ and make conjectures about where the maximum value occurs.

Possible approach

You need to find a local maximum for a small value of $n$ and then prove that this is the only maximum value. Clearly it is impossible to check all values of $n$. One method of proving the result uses the Binomial theorem.