Can the Product of All Primes Before p Be Greater Than p^2?

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In summary, The statement being proven is that for any prime number p greater than 11, the square of p will be smaller than the product of all primes that come before it. This is demonstrated with an example using p=13 and showing that 13^2 is less than the product of the first 4 primes (3*5*7*11). The person asking the question is wondering if this is generally true and if there is a simple proof available.
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cap.r
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Homework Statement


I am proving something different and need this to be true.

choose prime p > 11. then p^2 is less than the product of all primes that came before it.


Homework Equations


U(n)= {1, a_1, ... a_k} this is the ring of numbers co prime to n.

ex: let p=13. 13^2 = 169<3*5*7*11

The Attempt at a Solution



I am using 11 because it's not generally true for primes less than 11 and I have dealt with those cases in my proof.

is this generally correct? is there a simple proof I should show? or take it as general knowledge.
 
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  • #2
The "product of the first N primes" function grows so ridiculously fast as compared to the "square of the N-th prime" function, that pretty much any approximation at all should be usable to prove the inequality.
 

1. What are prime numbers?

Prime numbers are positive integers that are only divisible by 1 and themselves. They cannot be divided evenly by any other number.

2. How many prime numbers are there?

There are infinitely many prime numbers. As numbers get larger, it becomes increasingly difficult to determine if they are prime or not.

3. What is the definition of a prime power?

A prime power is a number that is the product of a prime number multiplied by itself a certain number of times. For example, 9 is a prime power because it is 3 to the power of 2.

4. Are all prime powers also prime numbers?

No, not all prime powers are also prime numbers. For example, 9 is a prime power, but it is not a prime number because it can be divided by 3.

5. How are prime numbers and prime powers used in mathematics?

Prime numbers and prime powers are used in fields such as number theory, cryptography, and coding theory. They also have practical applications in areas like computer science, finance, and physics.

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