No substitution please!

Number Theory Level 2

The no. of primes p such that p+1 is a perfect square.

If there are 5 primes ,then type the answer as 5


The answer is 1.

Bala Vidyadharan by Bala vidyadharan , 20, India

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1 solution

Ravi Dwivedi
Jul 17, 2015

p + 1 p+1 is a perfect square.

So let p + 1 = ( x + 1 ) 2 p+1=(x+1)^2 for some integer x x

⟹ p = x ( x + 2 ) \implies p=x(x+2)

{ x + 2 = p x = 1 ⟹ x = 1 , p = 3 \begin{cases}x+2=p\\ x=1\end{cases} \implies x=1,p=3

{ x + 2 = 1 x = p ⟹ x = − 1 , p = − 1 \begin{cases}x+2=1\\ x=p\end{cases} \implies x=-1,p=-1 i.e. no solutions.

These factors cannot be negative. So we have only one such prime p = 3 p=3

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Moderator note:

Simple standard approach, but the writeup leaves a lot to be desired. You could afford to be much clearer in expressing what you are trying to say. We are not mind-readers, and don't know what you are thinking.

Is there any reason why you chose = ( x + 1 ) 2 = (x+1)^2 as opposed to = x 2 = x^2 ?

Thought that 1 would cancel out but x 2 x^2 works the same. So both are same approaches

Ravi Dwivedi - 5 years, 11 months ago

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