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

Guest Jan 12, 2021

#1**0 **

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)

Arrakis Jan 13, 2021

#3**0 **

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

#5**+1 **

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.

Melody Jan 13, 2021

#7**+1 **

I shall try

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