# Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?

As odd squares are congruent to $$1$$ modulo $$8$$, any integer of the form $$4^k(8m+7)$$ with $$k,m\in\mathbb N=\{0,1,2,\ldots\}$$ cannot be written as the sum of three squares.

To avoid such congruence obstacles in representation problems, I suggest a variant of squares by using Euler's totient function $$\varphi$$. It is easy to see that all the numbers $$\varphi(n^2)=n\varphi(n)\ \ (n\in\mathbb Z^+=\{1,2,3,\ldots\})$$ are pairwise distinct.

QUESTION: Can we write each $$n\in\mathbb Z^+$$ as $$x^2+y^2+\varphi(z^2)$$ with $$x,y\in\mathbb N$$ and $$z\in\mathbb Z^+$$?

Actually, on Oct. 1, 2015, I even made the following stronger conjecture.

Conjecture. Any integer $$n>1$$ can be written as $$x^2+y^2+\varphi(z^2)$$, where $$x,y\in\mathbb N$$, $$x\le y$$, $$z\in\mathbb Z^+$$, and $$y$$ or $$z$$ is prime.

For the number of ways to write $$n$$ in this way, see http://oeis.org/A262311. For example, $$13$$ has a unique required representation: $$13=1^2+2^2+\varphi(4^2)$$ with $$2$$ prime, and $$94415$$ has a unique required representation: $$94415=115^2+178^2+\varphi(223^2)\ \ \text{with}\ 223\ \text{prime}.$$ I have verified the conjecture for all $$n=2,\ldots,10^6$$.

Though the conjecture might be quite challenging, the weaker version in the question should be on the research level. Any comments are welcome!

• Using the circle method it should be reasonably straightforward to show that the conjecture holds with z restricted to primes for almost all integers n > 1 and that the number of exceptions is << X^(1 - delta) for some positive delta > 0. – Mayank Pandey Nov 27 '18 at 23:30