There are three six-digit numbers such that each is a 4th power of its sum of digits .

1. List them.
2. Each one has a certain peculiarity (or more than one). Try to define it.
3. Ignoring the 6-digits constraint , how many integers like that exist?

DECLARE FUNCTION sod# (n#)
DEFDBL A-Z
FOR n = 0 TO 26873856 ' (8*9)^4
    s = sod(n)
    s = s * s * s * s
    IF s = n THEN PRINT n
NEXT n

FUNCTION sod (n)
st$ = LTRIM$(STR$(n))
t = 0
FOR i = 1 TO LEN(st$)
    t = t + VAL(MID$(st$, i, 1))
NEXT
sod = t
END FUNCTION

finds

0
1
2401
234256
390625
614656
1679616

By inspection you can see the three 6-digit solutions.

There can't be any more as the largest sod for an 8-digit number is 8*9; when raised to the 4th power, it's 26873856. Then (9*9)^4 and (10*9 )^4 are also only 8-digit numbers so no 9- or 10-digit number will work, nor any larger number as the number of digits in the power will always be smaller than the number of digits in the original number.

Posted by Charlie on 2012-08-14 15:24:35