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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?
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.khotsofalang said:is there any proof to show the non-existence of non-constant prime generating polynomial functions?
One way to prove that a polynomial function is not constant is by showing that the degree of the polynomial is greater than 0. If the degree is 0, then the function is a constant.
Yes, the Intermediate Value Theorem can be used to prove that a polynomial function is not constant. If the polynomial has at least one real root, then it must have at least one point where the function takes on a value between the values at the two roots. This means that the function is not constant.
Yes, there are a few ways to prove that a polynomial function is not constant without using calculus. These include using the Fundamental Theorem of Algebra, the Rational Root Theorem, and the Descartes' Rule of Signs.
Yes, a non-zero constant term can affect the non-constant nature of a polynomial function. For example, the function f(x) = x + 1 is not constant, even though it has a constant term of 1. However, a constant term of 0 would result in a constant function.
Yes, there are a couple of special cases where a polynomial function may appear to be non-constant, but is actually constant. One example is a polynomial with no real roots, such as f(x) = x^2 + 1. Another example is a polynomial with multiple roots, such as f(x) = (x-1)^2. In these cases, although the function is not constant, it only takes on one value for all inputs.