TITLE: Is it true that every positive integer is the sum of 18 fourth powers of integers?
QUESTION [5 upvotes]: Is it true that every positive integer is the sum of $18$ fourth powers of integers?
Does this means every positive integer $n = x_1^4+x_2^4+\cdots+x_n^4$ for some positive integer $n=18$? 
Could you show me some example, I don't think I am understanding the question..( ex:
Explanation says $78$ can be written as a sum of $18$ fourth powers of integers, how? Can $1$ or $5$ be written as sum of $18$ fourth powers of integers?)
Thanks!

REPLY [6 votes]: Note that some of the integers may be 0.  So $1 = 1^4 + 0^4 + \ldots + 0^4$.  $78 = 4 \times 2^4 + 14 \times 1^4$.  But what about 79?