Hint 1: A longer list:

• 1 = 1² + 0²
• 2 = 1² + 1²
• 3 (No)
• 4 = 2² + 0²
• 5 = 2² + 1²
• 6 (No)
• 7 (No)
• 8 = 2² + 2²
• 9 = 3² + 0²
• 10 = 3² + 1²
• 11 (No)
• 12 (No)
• 13 = 3² + 2²
• 14 (No)
• 15 (No)
• 16 = 4² + 0²
• 17 = 4² + 1²
• 18 = 3² + 3²
• 19 (No)
• 20 = 4² + 2²
• 21 (No)
• 22 (No)
• 23 (No)
• 24 (No)
• 25 = 5² + 0² = 4² + 3²
• 26 = 5² + 1²
• 27 (No)
• 28 (No)
• 29 = 5² + 2²
• 30 (No)

Hint 2: Consider the numbers that leave a remainder of 3 when you divide by 4.

Hint 3: If two numbers both work, then what about their product?