Proof for non constant polynomial function

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This discussion centers on the non-existence of non-constant prime generating polynomial functions. It references Euler's polynomial N^2 + N + 41, which generates primes for specific values of N but fails to do so universally. The fundamental proof asserts that no polynomial with integer coefficients can produce a prime for all integer inputs, as demonstrated by the divisibility of P(1 + t*p) by p when P(1) equals a prime. This proof extends to polynomials with rational coefficients as well.

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khotsofalang
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is there any proof to show the non-existence of non-constant prime generating polynomial functions?
 
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khotsofalang said:
is there any proof to show the non-existence of non-constant prime generating polynomial functions?
You got to be more specific about what you mean by a non-constant prime generating polynomial. If it is what I believe you mean, then this was noted in an earlier thread re Euler's function N^2 + N + 41. If you mean N takes only specific values such as "n = prime" or some sequence other than 1,2,3... then there is no such proof. If you omit the constant 41 then of course each integer will be composit for n > 1, however, the basic proof for non existence of polynominals in general (no polynomial with integer coefficients will generate a prime for all n since if P(1) = a prime "p" then P(1 + t*p) will always be divisible by p) will work whether there is or is not a constant in the polynomial such as 41.

Edit:I believe that a variation of the proof will work for polynomials with rational coefficients also.
 
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