1. ## urgent

Find all twin primes p, p+2 whose mid term p+1 is i) Triangular ii) Perfect square iii) perfect

thx

2. Originally Posted by marlen19861@hotmail.com

Find all twin primes p, p+2 whose mid term p+1 is i) Triangular ii) Perfect square iii) perfect

thx
i) If $p+1$ is a triangular number there exists a natural number $k$ such that:

$p+1=\frac{k(k+1)}{2}$

so:

$k^2+k-(2p+2)=0$

and hence:

$k=\frac{-1 \pm \sqrt{p+2}}{2}$

But the left hand side is an integer and the right hand side is irrational if $p+2$ is not a perfect square, but $p+2$ is prime. Hence there exist no twin priimes whose mid term is a triangular number.

CB

3. is the last question correct?

4. Originally Posted by nikolany
is the last question correct?
Now that I think about it it can't be since 5,6,7 would be a counter example.

The last equation should be:

$k=\frac{-1\pm\sqrt{8p+9}}{2}$

Then we require that $8p+9$ be a perfect square for both sides to be rational, which does not obviously work

CB

5. Originally Posted by CaptainBlack
Now that I think about it it can't be since 5,6,7 would be a counter example.

The last equation should be:

$k=\frac{-1\pm\sqrt{8p+9}}{2}$

Then we require that $8p+9$ be a perfect square for both sides to be rational, which does not obviously work

CB
Having thought about this some more I can now make this work, but I won't go into the detail of the solution as Opalg's solution in the other thread is much neater.

But the gist of the solution is that we can reduce the condition that $8p+9$ is a perfect square to: there exists a positive integer $k$ such that

$2p=(k-1)(k+2)$

which by the fundamental theorem of arithmetic forces us to conclude $p=??$.

CB