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

Age 16 to 18
Challenge Level

Show without recourse to any calculating aid that:

$$7^{1/2} + 7^{1/3} + 7^{1/4} < 7$$

and

$$4^{1/2} + 4^{1/3} + 4^{1/4} > 4$$

Sketch the graph of

$$f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x$$

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Shades of Fermat's Last Theorem

Exhaustion

Code to Zero

Power Up

Age 16 to 18Challenge Level

Age 16 to 18
Challenge Level