Irrationality of sum of roots of primes.

In summary, the conversation discusses the irrationality of various expressions involving square roots, including $\sqrt{2}$, $\sqrt{2}+\sqrt{3}$, and $\sqrt{2}+\sqrt{3}+\sqrt{5}$. It is suggested that $\sqrt{p_1} + \sqrt{p_2} + \sqrt{p_3} + \ldots + \sqrt{p_n}$ is also irrational for all $n$, where $p_i$ is the $i-th$ prime. However, the trick used to prove this for $n=3$ does not work for $n=4$.
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caffeinemachine
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MHB
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I observed the following:

1) $\sqrt{2}$ is irrational.

2) $\sqrt{2}+\sqrt{3}$ is irrational(since its square is irrational).

3) $\sqrt{2}+\sqrt{3}+\sqrt{5}$ is irrational(assume its rational and is equal to $r$. Write $r- \sqrt{5}=\sqrt{2} + \sqrt{3}$. Now square both the sides and its obvious from here).

So I am thinking may be $\sqrt{p_1} + \sqrt{p_2} + \sqrt{p_3} + \ldots + \sqrt{p_n}$ is irrational for all $n$, where $p_i$ is the $i-th$ prime.

The trick I used for (3) doesn't work for $n=4$. Any ideas anyone?
 
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FAQ: Irrationality of sum of roots of primes.

What is the irrationality of the sum of roots of primes?

The irrationality of the sum of roots of primes is a mathematical concept that states that the sum of the square roots of any two distinct prime numbers will always be irrational, meaning it cannot be expressed as a ratio of two integers. This has been proven by mathematicians using techniques such as proof by contradiction.

How does this concept relate to prime numbers?

This concept relates to prime numbers because it involves the sum of the square roots of two distinct prime numbers. Prime numbers are numbers that are only divisible by 1 and themselves, and they play a crucial role in this concept as the sum of their square roots is always irrational.

Can you give an example of the irrationality of the sum of roots of primes?

One example of the irrationality of the sum of roots of primes is the sum of the square roots of 2 and 3, which is approximately 3.14626437. This value cannot be expressed as a rational number and therefore confirms the concept of the irrationality of the sum of roots of primes.

What are the applications of this concept?

The irrationality of the sum of roots of primes has several applications in mathematics, including the proof of the infinitude of prime numbers and the proof of the irrationality of certain mathematical constants, such as pi and e. It also has practical applications in cryptography and number theory.

Is there a connection between the irrationality of the sum of roots of primes and the Goldbach's conjecture?

There is a conjecture that states that every even number greater than 2 can be written as the sum of two prime numbers. While this conjecture has not been proven, it is believed that the irrationality of the sum of roots of primes may have a connection to it. However, this connection has not been proven and remains an open question in mathematics.

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