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 is a triangular number there exists a natural number such that:
so:
and hence:
But the left hand side is an integer and the right hand side is irrational if is not a perfect square, but is prime. Hence there exist no twin priimes whose mid term is a triangular number.
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 is a perfect square to: there exists a positive integer such that
which by the fundamental theorem of arithmetic forces us to conclude .
CB