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.

Farey Neighbours

Age 16 to 18
Challenge Level

Start by constructing some Farey Sequences.

Find out more about the mediant of two fractions in Mediant Madness.

If you haven't yet met Proof by Induction, you may like to read this article.

For your proof by induction, your inductive hypothesis could be that if $\frac bd$ and $\frac ac$ are Farey Neighbours with $c,d \leq n$, then $ad-bc=1$.
Then consider which fractions can be Farey Neighbours of $\frac bd$ and $\frac ac$ in $F_{n+1}$.

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Shades of Fermat's Last Theorem

Exhaustion

Code to Zero

Farey Neighbours

Age 16 to 18Challenge Level

Age 16 to 18
Challenge Level