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is there any proof to show the non-existence of non-constant prime generating polynomial functions?

thats the proposition made by Euler that P(n)=n^2+n+41 can generate all primes and it was suddenly proven false by using a counter example, isn't there any way he can be proven to be false?

without using a counter example, show a proof that Euler's polynomial equation P(n)=n^2+n+41 can not be used to generate all the primes.can a general proof be made to show that all primes cannot be generated by a specific polynomial?

i need a solution to such an equation for stregthening my extended essay,anibody with a gud way of solving it?

all right, but what i actually need is that good way of solving it

i am sorry guys, the last time i posted this problem it was completely different but this time if we Let x12+x22=1 be a unit circle upon a finite field Zp where p is prime. Is there any algorithm which can give all the possible solutions (x1,x2) an element of Zp*Zp as well as the total number of...

Let x1 + x2 =1 be a unit circle upon a finite field Zp where p is prime. Is there any algorithm which can give all the possible solutions (x1,x2) an element of Zp*Zp as well as the total number of such solutions? If exists, what is the complexity of it?