For all three parts, consider integers with all of the following properties:
- the SOD(n^2) is also a square
- the SOD(n^3) is also a cube
- the SOD(n^4) is also a fourth power
- n is NOT a power of 10
Please find A, B, and C where:
(Part 1) A is the smallest such integer.
(Part 2) B is the smallest such integer whose first digit is different than the first digit of A.
(Part 3) C is the smallest such integer whose first digit is different than the first digit of A or B.
** SOD(n) is the Sum Of Digits of n
The great Euler conjected,
inter alia, that at least n powers of positive integers are needed to get a sum which is a n-th power as well.
It is up to you to find a set of integers {a,b,c,d,e}
such that
a^5+b^5+c^5+d^5=e^5.
REM: If unable to provide a proof, present the current state of known solutions for 4th and 5th powers.
AFAIK no solution for the 6th power exists so far.
