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Prove that 4 is the only perfect square that is one more than a prime number

 Jan 12, 2021
 #1
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I would say the answer is, No. I would say that because if you check the next perfect square, 8, it is one above 7 which is a prime number. (correct me if I'm wrong)

 Jan 13, 2021
 #2
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8 is not a perfect square.

Guest Jan 13, 2021
 #3
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I am very sorry. I am a bit rusty on my elementary math... it has been about a year since I've had a refresher for it.  Let me fix that mistake and take a bit of time into that. I would say it is the only perfect square that is one above a prime number I went all the way to 263.

Arrakis  Jan 13, 2021
 #4
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It's not a yes or no answer it's telling to prove how, and I should add that it's supposed to use factoring.

Guest Jan 13, 2021
 #5
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A perfect square is of the form x^2 where x is a positive whole number

One less than a perfect square is    x^2 -1

 

\(x^2-1=(x-1)(x+1)\)

 

If this is a prime number then the smaller factor must be 1

x-1=1

x=2

x^2-1 = 3

Hence 3 is the only prime number that is followed by a perfect square.

 Jan 13, 2021
 #6
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Can you explain how you got to x-1=1

Guest Jan 14, 2021
 #7
avatar+112032 
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I shall try  smiley

 

A perfect square is of the form x^2 where x is a positive whole number

One less than a perfect square is    x^2 -1

 

\(x^2 -1\)     is the difference of 2 squares.

 

difference means subtract and the two squares are   x^2    and    1^2

You need to memorize this factorization.

 

  \(\boxed{a^2-b^2=(a-b)(a+b)}\)

 

so        \(x^2-1^2=(x-1)(x+1)\)

 

If this is a prime number then the smaller factor must be 1

the factors are  (x-1) and (x+1)

since x is a positive whole number,  the smallest one will be x-1

therefore  x-1 must equal 1

so x must equal 2

Melody  Jan 15, 2021

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