can you please help?
Find all twin primes p, p+2 whose mid term p+1 is i) Triangular ii) Perfect square iii) perfect
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.
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 .