Inequalities #2
Find a four-digit number such that it equals to the forth power of the sum of all of its integers

Question

Inequalities #2 
Find a four-digit number such that it equals to the forth power of the sum of all of its integers

anonymous · Answer

2401.

anonymous · Answer

Method I used: look at fourth powers with four digits, there are only a few.

callisto · Answer

Yup... there are only a few... but how to do it (not by trial and error) ?

anonymous · Answer

$$
1000w+100x+10y+z=n^4\
w+x+y+z=n\
n\in\mathbb{N}
$$
Maybe. Dunno.

experimentx · Answer

1000a+100b+10c+d = (a+b+c+d)^4
5< a+b+c+d < 10

anonymous · Answer

1634
8208
9474

anonymous · Answer

the forth power of the sum of all of its integers:(
i am sorry! i find the  the sum of the forth power of all of its integers

callisto · Answer

Never mind. :)
So... it seems easier now.. but still by trial and error :(

callisto · Answer

Hmm... I wish I could take back my words...
Actually, solving it by trial and error is not the problem. But I wish to find a better method only. 
Thanks all :)

anonymous · Answer

9999^1/4<10
1000^1/4>5
So, a+b+c+d=(6,7,8,9)
6^4=1296
7^4=2401
8^4=4096
9^4=6561

a=2
b=4
c=0
d=1