$a^2 + b^2 + c^2 + d^2 + e^2=f^2$

Question

I don't mean to ask for the solution but there is a challenging word problem that I have no idea how to start and would appreciate some help thanks.

What is the smallest value of $f$ that satisfies $a^2 + b^2 + c^2 + d^2 + e^2=f^2$?

Answer 1

Hint (assuming $a,b,c,d,e,f$ must be positive integers): Can you get $f=1$? $2$? $3$? $4$?

Answer 2

If $0$ is allowed, the answer is $0$. If all the variables have to be positive integers the left side is at least $5$ and $9$ doesn't work. If you can do $16$ you are done.

Answer 3

the smallest square that is the sum of five distinct positive squares is $$ 10^2 = 7^2 + 5^2 + 4^2 + 3^2 + 1^2 $$

The only square that is the sum of more than one consecutive positive squares, beginning with $1,$ is $$ 70^2 = 1^2 + 2^2 + 3^2 + 4^2 + \cdots + 24^2 \; . $$ This leads to one construction of the Leech Lattice. See page 130 in the second edition of Lattices and Codes by Wolfgang Ebeling. The raw fact is proved in Mordell's book, Diophantine Equations.