Question

If p is a prime less than 500, how many number of the form 3p + 1 are perfect squares?

Answer 1

Anybody??????

Answer 2

One.

Answer 3

\[ p=5; 3p+1 = 16 = 4^2 \]

Answer 4

A perfect square ends with 00,4,5,6,9
so p must have its unit digit as 1 or 5 and then i substituted the value

Answer 5

\(3p+1 = k^2 \) (say)

\[ 3p = (k+1)(k-1) \]

Now there will be two cases either (k+1)=3 and (k-1) = p or viceversa.

Answer 6

can you do the rest?

Answer 7

Anyways,

If \( (k+1)=3 \) and \( (k-1) = p \implies p=1 \) which is not prime

If \( (k+1)=p \) and \( (k-1) = 3 \implies p=5 \implies 3p+1 = 16 = 4^2 \)

There will be only one such prime.

Q.E.D

Answer 8

Oh yes