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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.

# Power Up

##### Stage: 5 Challenge Level:

As a hint, try comparing the '$7$'s' inequality to a similar one for $8$. For the '$4$'s' inequality use the fact that any root of $4$ is greater than $1$.

To sketch the graph, find the derivative for $x=0$ and then consider where the derivative is positive, where it is negative and if it tends to a limit as $x$ increases.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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