Let $\mathbb N=\{0,1,2,\ldots\}$. Those $T_n:=n(n+1)/2$ with $n\in\mathbb N$ are called triangular numbers. It is well known that $$\{T_a+T_b+T_c:\ a,b,c\in\mathbb N\}=\mathbb N\tag{1}$$ which was conjectured by Fermat and proved by Gauss. Euler observed that $$\begin{aligned}&\{(2x)^2+(2y+1)^2=4(x^2+2T_y)+1:\ x,y\in\mathbb N\} \\=&\{(a-b)^2+(a+b+1)^2=4(T_a+T_b)+1:\ a,b\in\mathbb N\}.\end{aligned}\tag{2}$$

Motivated by $(1)$ and $(2)$, I formulated the following conjecture on June 7, 2019.

**Conjecture.** Let $n$ be any positive integer. Then we can write $n$ as $T_a+T_b+4^c5^d$ with $a,b,c,d\in\mathbb N$; equivalently, $n$ can be written as $u^2+2T_v+4^x5^y$ with $u,v,x,y\in\mathbb N$ (or $4n+1=a^2+b^2+4^c5^d$ for some $a,b,c,d\in\mathbb N$ with $c>0$).

I have verified this for all $n=1,\ldots,5\times10^8$. For example, $585$ has a unique required representation: $$585 = T_{10}+T_{20}+4^35^1=5^2 +2T_{15} + 4^35^1.$$

**QUESTION.** Can one find a counterexample to the conjecture?