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mathworker
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is there any way to prove or disprove the statement:
y=3x^2+3x+1 is prime for all x belongs to natural numbers...
y=3x^2+3x+1 is prime for all x belongs to natural numbers...
mathworker said:is there any way to prove or disprove the statement:
y=3x^2+3x+1 is prime for all x belongs to natural numbers...
mathworker said:is there any way to prove or disprove the statement:
y=3x^2+3x+1 is prime for all x belongs to natural numbers...
chisigma said:Is...
$$ y = 3\ (x-x_{1})\ (x-x_{2})\ (1)$$
... where $x_{1}$ and $x_{2}$ are the roots of the equation $x^{2} + x + \frac{1}{3} = 0$. The discriminat is <0, so that no real roots exists and that means that no real factor of y in (1) exists...
Kind regards
$\chi$ $\sigma$
chisigma said:Is...
$$ y = 3\ (x-x_{1})\ (x-x_{2})\ (1)$$
... where $x_{1}$ and $x_{2}$ are the roots of the equation $x^{2} + x + \frac{1}{3} = 0$. The discriminant is <0, so that no real roots exists and that means that no real factor of y in (1) exists...
Kind regards
$\chi$ $\sigma$
Fernando Revilla said:Yes, use the following theorem (Goldbach 1752):
If $f\in\mathbb{Z}[x]$ has the property that $f(n)$ is prime for all $n\ge 1$, then $f$ must be a constant.
mathworker said:but given function is not constant and i checked y =f(X) for some numbers and found it prime every time
mathworker said:i checked y =f(X) for some numbers and found it prime every time
Fernando Revilla said:For some, but not for all. :)
Proving the primality of a quadratic over the natural numbers means to show that the quadratic equation has no other factors besides 1 and itself when the input values are restricted to natural numbers. In other words, it is a way to determine if a quadratic equation is a prime number when using only whole numbers as inputs.
There are several methods to prove the primality of a quadratic over the natural numbers, such as trial division or using the Sieve of Eratosthenes. However, for larger numbers, more sophisticated algorithms like the Miller-Rabin primality test or the AKS primality test are used.
Proving the primality of a quadratic over the natural numbers is important in number theory and cryptography. It is used to determine the security of cryptographic algorithms and to understand the distribution of prime numbers in a given range.
One of the main challenges is the time and computational resources required, especially for larger numbers. Additionally, the existence of efficient factorization algorithms can make it difficult to prove the primality of a quadratic over the natural numbers.
Yes, there are many real-world applications, including in cryptography for secure communication and data encryption. It is also used in the development of secure digital signatures and in the generation of random numbers for various applications. Additionally, it has implications in fields such as physics, biology, and economics.