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?

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.

Challenge Level

Find the range of values of $x$ for which

$$

{\sqrt{x}+ {{1}\over{\sqrt{x}}}} < {4}\,,

$$

where $\sqrt{x}$ is the positive root.

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NRICH is part of the family of activities in the Millennium Mathematics Project.