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16 April, 07:22

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

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  1. 16 April, 07:50
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    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.
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