In this problem, Alex is thinking of two numbers that satisfy two conditions:

1. When added together, they form a square number.
2. When multiplied, they give a prime number.

Step 1: Analyze the conditions

• Sum of the numbers = square number: This means that the sum of the two numbers must be a perfect square (e.g., 1, 4, 9, 16, 25, etc.).

• Product of the numbers = prime number: A prime number is a number that has only two distinct divisors, 1 and itself. The product of the two numbers must result in a prime number.

Step 2: Check small numbers

Since the product of the two numbers is a prime number, one of the numbers must be 1, because multiplying any number by 1 results in that number. The other number must then be the prime number itself.

Case 1: Let one of the numbers be 1

Now, check small prime numbers to see if their sum with 1 is a perfect square.

• Prime number = 2:

  • Sum = $1 + 2 = 3$ (not a square number)

• Prime number = 3:

  • Sum = $1 + 3 = 4$ (this is a square number, $2^2$)

So, the numbers are 1 and 3. They satisfy both conditions:

• Sum: $1 + 3 = 4$ (a perfect square).
• Product: $1 \times 3 = 3$ (a prime number).

Conclusion:

Alex could be thinking of the numbers 1 and 3.

Would you like more details or have any questions on this? Here are 5 related questions:

1. What are some other examples of pairs where the sum is a perfect square and the product is a prime number?
2. Why must one of the numbers be 1 in this case?
3. Can prime numbers be negative in this context?
4. What are the properties of prime numbers relevant to this problem?
5. What other mathematical scenarios require the sum to be a perfect square?

Tip: When dealing with prime number products, always consider 1 as a possible factor since it simplifies the process!