Can a sum of $n$ squares be expressed as the sum of $n/2$ squares?

The answer is yes for (even) $n \geq 8$ and no for (even) $n \leq 7$.

If $n \geq 8$ then the sum of your $n$ squares is the sum of four squares by the Lagrange four square theorem. Now, if $n/2$ is greater than 4, you can complete your sum by adding enough terms equal to $0^2$.

For $4 \leq n \leq 7$ note that $7$ can be written as the sum of $n$ squares but cannot be written as the sum of $n/2$ squares.

For $2 \leq n \leq 3$ note that $5$ is the sum of $n$ squares but not the sum of $n/2$ squares.

Tags:

Sums Of Squares

Pythagorean Triples

Elementary Number Theory

Related

For a positive integer $n\geq 2$ with divisors $1=d_1<d_2<\cdots<d_k=n$, prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k<n^2$ Suppose $\angle BAC = 60^\circ$ and $\angle ABC = 20^\circ$. A point $E$ inside $ABC$ satisfies $\angle EAB=20^\circ$ and $\angle ECB=30^\circ$. Does a nontrivial finite solvable group have a subgroup of prime power index for each prime divisor? If $V_n(a)$ counts sign changes in the sequence $\cos a, \cos2a,\cos3a,\ldots,\cos na,$ show that $\lim_{n\to\infty}\frac{V_n(a)}n=\frac{a}\pi$ probability group students to maximize expectation Is there an efficient way of showing $\int_{-1}^{1} \ln\left(\frac{2(1+\sqrt{1-x^2})}{1+x^2}\right)dx = 2$? Difficulty in understanding significance of Grelling's Paradox. $0 \leq u - \ln(1+u) \leq u^2$ for $\mid u\mid < \frac{1}{2}$ Why doesn't the fact we can force the continuum hypothesis outright prove the continuum hypothesis? Olympiads Number Theory Problem: Show that there are no positive $a,b,c\in\Bbb Z$ such that $\frac{a^2+b^2+c^2}{3(ab+bc+ca)}\in\Bbb Z$ Is there a bijective function $f:[0,1] \to [0,1]$ such that the graph of $f$ in $\mathbb{R}^2$ is a dense subset of $[0,1] \times [0,1]$? Degree of a field extension by a transcendental element

Recent Posts