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.

Farey Neighbours

Age 16 to 18 Challenge Level:

We received a very clear solution from Amrit, proving using induction that for any Farey Neighbours, $ad-bc=1$.
Here is Amrit's Solution.