Antoine has three tomato plants. The total number of tomatoes on all his plants is \(41\), and one of his plants has exactly \(4\) more tomatoes than another plant. If the number of tomatoes on each plant is a prime number, determine all possibilities for the number of tomatoes on each plant.
Note: A prime number is an integer greater than \(1\) that has only two positive factors: \(1\) and itself. The number \(17\) is prime because its only positive factors are \(1\) and \(17\).
This problem is asking us to find three prime numbers with a sum of \(41\), where two of the prime numbers differ by exactly \(4\). We can start by listing all the prime numbers less than \(41\). They are \(2\), \(3\), \(5\), \(7\), \(11\), \(13\), \(17\), \(19\), \(23\), \(29\), \(31\), and \(37\).
Next we can look for all pairs of numbers from this list that differ by \(4\). These pairs are as follows: \(3\) and \(7\), \(7\) and \(11\), \(13\) and \(17\), and \(19\) and \(23\).
Since we want three prime numbers that add to \(41\), we can look at each of these pairs of numbers to determine all possible solutions.
\(3\) and \(7\): The third number would be \(41-3-7=31\). Since this is a prime number, this is a possible solution.
\(7\) and \(11\): The third number would be \(41-7-11=23\). Since this is a prime number, this is a possible solution.
\(13\) and \(17\): The third number would be \(41-13-17=11\). Since this is a prime number, this is a possible solution.
\(19\) and \(23\): Since \(19+23=42\), this sum is already over \(41\), so this is not a possible solution.
Therefore, there are three solutions. The tomato plants could have \(3\), \(7\), and \(31\) tomatoes each, \(7\), \(11\), and \(23\) tomatoes each, or \(13\), \(17\), and \(11\) tomatoes each.