Is every integer the sum of distinct prime numbers?

Let $\Bbb{P}$ be the set of prime numbers and $Q=\Bbb{P}\cup\{1\}$. Is it true that every natural numbers ($\neq 0$) is the sum of distinct elements of $Q$? I tried from $1$ to $60$ and it seems true.

Solution 1:

By Bertrand's postulate, you can find a prime satisfying $\lfloor n/2\rfloor <p< n$. Proceed by induction.

Solution 2:

Note that we can not write $2$ in required form.

Take any integer (say) $x$ with $3\le x.$
Let $y$ be the largest prime strictly less than $x.$ If $x-y\in Q$ we are already done.
Suppose $x-y\not\in Q.$ Then, take the largest prime strictly less than $x-y.$
And continue this..