4. Represent the number 43 as the sum of three prime numbers. In how many ways can this be done? Note that the number 1 is not considered a prime number.
Solution.

$$
43=a+b+c, a, b, c \text { - prime numbers. }
$$

Prime numbers from 2 to 43

| 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |

$a=2 \rightarrow 43-4=41 \rightarrow b=2 \rightarrow c=39-$ not a prime number.

$a>2$ : list $a, 43-a$ and prime numbers $b, c$, for which $43-a=b+c$.

| $a$ | 3 | 3 | 3 | 5 | 5 | 7 | 7 | 7 | 7 | 11 | 11 | 13 | 13 | 13 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $43-a$ | 40 | 40 | 40 | 38 | 38 | 36 | 36 | 36 | 36 | 32 | 32 | 30 | 30 | 30 |
| $b$ | 3 | 11 | 17 | 7 | 19 | 5 | 7 | 13 | 17 | 3 | 13 | 7 | 11 | 17 |
| $c$ | 37 | 29 | 23 | 31 | 19 | 31 | 29 | 23 | 19 | 29 | 19 | 23 | 19 | 13 |


| $a$ | 17 | 17 | 17 | 19 | 19 | 19 | 23 | 23 | 29 | 29 | 31 | 37 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $43-a$ | 26 | 26 | 26 | 24 | 24 | 24 | 20 | 20 | 14 | 14 | 12 | 6 |
| $b$ | 3 | 7 | 13 | 5 | 7 | 11 | 3 | 7 | 3 | 7 | 5 | 3 |
| $c$ | 23 | 19 | 13 | 19 | 17 | 13 | 17 | 13 | 11 | 7 | 7 | 3 |

Different triples of prime numbers, the sum of which is 43:

$$
\begin{aligned}
& 3+3+37,3+11+29,3+17+23,5+7+31,5+19+19,7+7+29, \\
& 7+13+23,7+17+19,11+13+19,13+17+13
\end{aligned}
$$

Answer: ten ways.