What prime is 4 greater than a perfect square and 7 less than the next-largest perfect square?

X - the prime

y² - the first perfect square

(y + 1) ² - the second perfect square

(1) x = y² + 4

(2) x = (y + 1) ² - 7

subtitute (2) to (1)

(y + 1) ² - 7 = y² + 4 |use (a + b) ² = a² + 2ab + b²

y² + 2y + 1 - 7 = y² + 4 |subtract y² from both sides

2y - 6 = 4 |add 6 to both sides

2y = 10 |divide both sides by 2

y = 5

subtitute the value of y to (1)

x = 5² + 4 = 25 + 4 = 29

check:

29 - it's prime / correct/

29 - 4 = 25 = 5² - the perfect square / correct/

29 + 7 = 36 = 6² - the next-largest perfect square / correct/

Answer: 29.