If anyone of you think this is ( the $n = 4$ case) just the Lagrange's four square theorem, you're wrong. The four square theorem does not include the word uniquely !!! So any ideas now!??!???!??!

Note that 5 = 1+4 I think you can complete the proof from here....

Rushil wrote: Find all positive integers $n$ that can be uniquely expressed as a sum of five or fewer squares. Anyone for this interesting question?

Rushil wrote: Find all positive integers $n$ that can be uniquely expressed as a sum of five or fewer squares. Let $(l_1(n),l_2(n),l_3(n),l_4(n))$ a four square Lagrange expression for $n$ (such that $n=l_1(n)^2+l_2(n)^2+l_3(n)^2+l_4(n)^2$ If $n\ge 16$, we have at least 5 expressions of $n$ in sum of 5 squares : $n=0^2+l_1(n)^2+l_2(n)^2+l_3(n)^2+l_4(n)^2$ $n=1^2+l_1(n-1)^2+l_2(n-1)^2+l_3(n-1)^2+l_4(n-1)^2$ $n=2^2+l_1(n-4)^2+l_2(n-4)^2+l_3(n-4)^2+l_4(n-4)^2$ $n=3^2+l_1(n-9)^2+l_2(n-9)^2+l_3(n-9)^2+l_4(n-9)^2$ $n=4^2+l_1(n-16)^2+l_2(n-16)^2+l_3(n-16)^2+l_4(n-16)^2$ And these 5 expressions can be all the same only if they are $0^2+1^2+2^2+3^2+4^2=30$ And since $30=1^2+2^2+3^2+4^2=1^2+2^2+5^2$, no number $\ge 16$ fit the requirement and we just have to check numbers from $1$ to $15$ : $15=3^2+2^2+1^2+1^2$ has a unique expression $14=3^2+2^2+1^2=2^2+2^2+2^2+1^2+1^2$ has at least two expressions $13=3^2+2^2=2^2+2^2+2^2+1^2$ has at least two expressions $12=3^2+1^2+1^2+1^2=2^2+2^2+2^2$ has at least two expressions $11=3^2+1^1+1^2=2^2+2^2+1^2+1^2+1^2$ has at least two expressions $10=3^2+1^1=2^2+2^2+1^2+1^2$ has at least two expressions $9=3^2=2^2+2^2+1^2$ has at least two expressions $8=2^2+2^2=2^2+1^2+1^2+1^2+1^2$ has at least two expressions $7=2^2+1^2+1^2+1^2$ has a unique expression $6=2^2+1^2+1^2$ has a unique expression $5=2^2+1^2=1^2+1^2+1^2+1^2+1^2$ has at least two expressions $4=2^2=1^2+1^2+1^2+1^2$ has at least two expressions $3=1^2+1^2+1^2$ has a unique expression $2=1^2+1^2$ has a unique expression $1=1^2$ has a unique expression Hence the answer : $\boxed{\{1,2,3,6,7,15\}}$