can you please help?
Find all twin primes p, p+2 whose mid term p+1 is i) Triangular ii) Perfect square iii) perfect
thx
i) If $\displaystyle p+1$ is a triangular number there exists a natural number $\displaystyle k$ such that:
$\displaystyle p+1=\frac{k(k+1)}{2}$
so:
$\displaystyle k^2+k-(2p+2)=0$
and hence:
$\displaystyle k=\frac{-1 \pm \sqrt{p+2}}{2}$
But the left hand side is an integer and the right hand side is irrational if $\displaystyle p+2$ is not a perfect square, but $\displaystyle p+2$ is prime. Hence there exist no twin priimes whose mid term is a triangular number.
CB
Now that I think about it it can't be since 5,6,7 would be a counter example.
The last equation should be:
$\displaystyle k=\frac{-1\pm\sqrt{8p+9}}{2}$
Then we require that $\displaystyle 8p+9$ 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 $\displaystyle 8p+9$ is a perfect square to: there exists a positive integer $\displaystyle k$ such that
$\displaystyle 2p=(k-1)(k+2)$
which by the fundamental theorem of arithmetic forces us to conclude $\displaystyle p=??$.
CB