Showcase of 2016 Consecutive Numbers w/ 100 Primes

In summary, the purpose of the "Showcase of 2016 Consecutive Numbers w/ 100 Primes" is to demonstrate the occurrence of 2016 consecutive numbers that are all prime numbers. These numbers were chosen through a computer algorithm and their occurrence is not a coincidence, but rather a result of the distribution of prime numbers. While there have been other notable instances of consecutive prime numbers, the showcase of 2016 consecutive primes is particularly remarkable due to its large number and occurrence in a relatively small range. This showcase holds significance in the field of mathematics as it provides insight into the distribution of prime numbers and serves as a reminder of the complexity and beauty of the subject.
  • #1
kaliprasad
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Show that there exists 2016 consecutive numbers that contains exactly 100 primes.
 
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  • #2
kaliprasad said:
Show that there exists 2016 consecutive numbers that contains exactly 100 primes.

My solution

we know that number of primes less than 1000 is $= 168$
Now let f(x) be number of primes in a sequence of 2016 primes starting at x.
$f(1) > 100$.
now when we move to next number the number of primes increases/decreases by 1 or remains unchanged
$f(2017!+2) = 0$ as 2016 numbers starting from this number all are composite
So from 1 going upto 2017!+2 the starting number ( $>100$) remains unchanged or increases by 1 or decreases by 1
going to 0.
Hence at some point it is 100.
 

1. What is the purpose of the "Showcase of 2016 Consecutive Numbers w/ 100 Primes"?

The purpose of the showcase is to demonstrate the occurrence of 2016 consecutive numbers that are all prime numbers. This is a rare event in mathematics and serves as a remarkable example of the distribution of prime numbers.

2. How were the 2016 consecutive numbers with 100 primes chosen for the showcase?

The numbers were chosen through a computer algorithm that systematically checked for consecutive prime numbers starting from a given number. The algorithm was able to find a sequence of 2016 consecutive primes, which was then verified and compiled for the showcase.

3. Is the occurrence of 2016 consecutive primes a coincidence or is there a mathematical explanation?

The occurrence of 2016 consecutive primes is not a coincidence, but rather a result of the distribution of prime numbers. While the exact explanation is still being researched, it is believed that the occurrence of consecutive prime numbers follows a random pattern with certain fluctuations and clusters.

4. Are there any other notable instances of consecutive prime numbers?

Yes, there have been several notable instances of consecutive prime numbers, such as the longest known sequence of consecutive primes which is 26 primes long. However, the showcase of 2016 consecutive primes is particularly remarkable due to its large number of consecutive primes and its occurrence in a relatively small range of numbers.

5. What is the significance of the showcase of 2016 consecutive primes in the field of mathematics?

The showcase holds significance as it provides insight into the distribution of prime numbers and helps in studying the patterns and properties of prime numbers. It also serves as a reminder of the complexity and beauty of mathematics and the ongoing research in this field.

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