Finding the Minimum $x$ for a Prime Product

In summary, the "minimum $x$ for a prime product" problem involves finding the smallest positive integer $x$ for which the product of two prime numbers is equal to $x$. It is relevant in science as prime numbers are crucial in many areas of mathematics and there is a known solution to this problem known as "Goldbach's Conjecture". While it may not have direct applications in real life, the study of prime numbers has led to important technologies and there is ongoing research on this problem.
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anemone
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Find the minimum value for $x$ where $x\in \mathbb{N} $ and $x^2-x+11$ can be written as a product of 4 primes, which are not necessarily distinct.
 
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The key question to ask is: which primes can occur as factors of $N = x^2 - x + 11$, where $x$ is an integer?
If $x$ is an integer then either $x$ or $x-1$ is even, so $x^2-x = x(x-1)$ is always even and therefore $N = x^2 - x + 11$ is always odd. Thus the prime $2$ can never be a divisor of $N$.

Can the prime $3$ occur as a divisor of $N$? If so then (working mod $3$) $$0 \equiv x^2 - x + 11 \equiv x^2 + 2x + 2 \equiv (x+1)^2 + 1 \pmod3.$$ Therefore $(x+1)^2 \equiv -1\equiv 2 \pmod3$. But that equation has no solutions. In the language of number theory, $2$ is not a quadratic residue mod $3$. Therefore $3$ can never be a prime factor of $N$.

Can the prime $5$ occur as a divisor of $N$? If so then $0 \equiv x^2 - x + 11 \equiv x^2 + 4x + 1 \equiv (x+2)^2 + 2 \pmod5$. Therefore $(x+2)^2\equiv -2\equiv3 \pmod5$. But $3$ is not a quadratic residue mod $5$. Therefore $5$ can never be a prime factor of $N$.

Can the prime $7$ occur as a divisor of $N$? If so, then $0 \equiv x^2 - x + 11 \equiv x^2 + 6x + 4 \equiv (x+3)^2 + 2 \pmod7$. Therefore $(x+3)^2\equiv -2\equiv5 \pmod7$. But $3$ is not a quadratic residue mod $7$. Therefore $7$ can never be a prime factor of $N$.

But the prime $11$ can occur as a divisor of $N$. So can the prime $13$. So the smallest possible candidate for $N$ to be a product of four primes would be if $N = 11^4 = 14641$. However, if $x = 121$ then $x^2 - x + 11 = 121*120 + 11 = 14531 < 14641$, but if $x = 122$ then $x^2 - x + 11 = 122*121 + 11 = 14773 > 14641$. So $11^4$ falls between two values of $x^2 - x + 11$ and therefore does not give a solution to the problem.

The next smallest possible candidate for $N$ is $11^3*13 = 17303$. This time we strike lucky, because if $x=132$ then $x^2 - x + 11 = 132*131 + 11 = 17303$. So the minimum value for $x$ such that $x^2 - x + 11$ is a product of four primes is $x = 132$, and the primes are then $11,11,11,13$.
 

FAQ: Finding the Minimum $x$ for a Prime Product

What is the "Finding the Minimum $x$ for a Prime Product" problem?

The "Finding the Minimum $x$ for a Prime Product" problem is a mathematical problem that involves finding the smallest positive integer $x$ such that the product of all the prime numbers from 1 to $x$ is greater than or equal to a given number $n$.

How is this problem relevant in real life?

This problem has applications in cryptography and computer science, as it is used to generate large prime numbers for secure encryption. It is also used in economics and finance to calculate the minimum number of units of a product that need to be produced to meet a certain demand.

What is the most efficient way to solve this problem?

The most efficient way to solve this problem is to use a computer program or algorithm that can quickly calculate the product of all the prime numbers from 1 to $x$ and compare it to the given number $n$. This approach is much faster than manually calculating the product and checking for the minimum $x$.

Are there any known solutions for this problem?

Yes, there are known solutions for this problem. One of the most well-known is the Bertrand's postulate, which states that for any positive integer $n$, there exists at least one prime number between $n$ and $2n$. This can be used to find the minimum $x$ for a prime product by repeatedly multiplying the prime numbers within this range until the product is greater than or equal to $n$.

Is there a limit to the size of $x$ in this problem?

Yes, there is a limit to the size of $x$ in this problem. As the product of prime numbers grows exponentially, the value of $x$ will eventually become too large to be represented by a computer. This limit is dependent on the computing power and storage capacity available.

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