Discovering Intersections of Infinite Primes Sets

In summary, there are many different subsets of primes including real Eisenstein primes, Pythagorean primes, real Gaussian primes, Landau primes, factorial primes, and more. It is possible to find numbers that belong to multiple sets, and it is also possible to compare the number of intersections between different pairs of sets. However, because primes are highly irregular, it may be difficult to determine the behavior of the intersections of the entire sets.
  • #1
caters
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We have this set of primes which is infinite. This has lots of different subsets. Here is the list of subsets:
Real Eisenstein primes: 3x + 2
Pythagorean primes: 4x + 1
Real Gaussian primes: 4x + 3
Landau primes: x^2 + 1
Central polygonal primes: x^2 - x + 1
Centered triangular primes: 1/2(3x^2 + 3x + 2)
Centered square primes: 1/2(4x^2 + 4x + 2)
Centered pentagonal primes: 1/2(5x^2 + 5x + 2)
Centered hexagonal primes: 1/2(6x^2 + 6x + 2)
Centered heptagonal primes: 1/2(7x^2 + 7x + 2)
Centered decagonal primes: 1/2(10x^2 + 10x + 2)
Cuban primes: 3x^2 + 6x + 4
Star Primes: 6x^2 - 6x + 1
Cubic primes: x^3 + 2
Wagstaff primes: 1/3(2^n + 1)
Mersennes: 2^x - 1
thabit primes: 3 * 2^x - 1
Cullen primes: x * 2^x + 1
Woodall primes: x * 2^x - 1
Double Mersennes: 1/2 * 2^2^x - 1
Fermat primes: 2^2^x + 1
Alternating Factorial Primes: if x! has x being odd than every odd number when you take the factorial positive and every even number negative. Opposite for even indexed factorials. For example 3rd alternating factorial = 1! - 2! + 3!
Primorial primes: First n primes multiplied together - 1
Euclid primes: first n primes multiplied together + 1
Factorial primes: x! + 1 or x! - 1
Leyland primes: m^n + n^m where m can be anything not negative but n has to be greater than 1
Pierpont primes: 2^m * 3^n + 1
Proth primes: n * 2^m + 1 where n < 2^m
Quartan primes: m^4 + n^4
Solinas primes: 2^m ± 2^n ± 1 where 0< n< m
Soundararajan primes: 1^1 + 2^2 ... n^n for any n
Three-square primes: l^2 + m^2 + n^2
Two Square Primes: m^2 + n^2
Twin Primes: x, x+2
Cousin primes: x, x+4
Sexy primes: x, x + 6
Prime triplets: x, x+2, x+6 or x, x+4, x+6
Prime Quadruplets: x, x+2, x+6, x+8
Titanic Primes: x > 10^999
Gigantic Primes: x > 10^9999
Megaprimes: x > 10^999999

Now Here is a question. Can you find a number where at least 2 of the sets intersect? I will try to do this myself. Just so you know I am going up to 12 digit primes because that is the largest prime my computer will test without the program taking too long to test it and to be sure I find intersections of the sets.

Another question is if I use the number of intersections between 2 sets of 2 types of primes up to 12 digits and I compare that to the number of intersections in a different pair of sets can I figure out which sets have the most intersections?

I am in 9th grade but know some trig and precalculus.
 
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  • #2
caters said:
Now Here is a question. Can you find a number where at least 2 of the sets intersect? I will try to do this myself. Just so you know I am going up to 12 digit primes because that is the largest prime my computer will test without the program taking too long to test it and to be sure I find intersections of the sets.

Well, yeah, it's trivial. 5 is a "real eisenstein prime", a pythagorean prime, a landau prime, a fermat prime, a factorial prime, it has two twin primes (3 and 7), and so on... there are infinitely many primes that are titanic primes, gigantic primes, and megaprimes (since all megaprimes are both titanic and gigantic, and there are infinitely many of them), etc... many of these sets overlap, some of them significantly (e.g. eisenstein and gaussian primes).

caters said:
Another question is if I use the number of intersections between 2 sets of 2 types of primes up to 12 digits and I compare that to the number of intersections in a different pair of sets can I figure out which sets have the most intersections?

I don't know that an exhaustive test to check how much overlap any two sets have up to 12 digits is going to tell you very much about the behaviour of the intersection of the whole sets. Primes are highly irregular, it's not even known that a lot of these subsets are actually infinite. In fact I'm pretty sure for most of these subsets getting any (mathematical) upper bound on the number of intersections below a certain prime is going to be difficult, and a few could probably even be considered open problems.

But yes, if you restrict yourself to primes up to 12 digits, then you can work out the intersection of each set with every other, and then use the inclusion exclusion principle to figure out which sets intersect the others the most according to some metric of your choosing.
 

FAQ: Discovering Intersections of Infinite Primes Sets

What is the concept of "Discovering Intersections of Infinite Primes Sets"?

The concept of "Discovering Intersections of Infinite Primes Sets" involves analyzing the intersections between different sets of infinite prime numbers. This can help us gain a deeper understanding of the distribution and patterns of prime numbers.

Why is it important to study the intersections of infinite primes sets?

Studying the intersections of infinite primes sets can provide valuable insights into the properties and behavior of prime numbers. This can have implications in various fields, such as cryptography, number theory, and computer science.

How do scientists discover intersections of infinite primes sets?

Scientists use various mathematical techniques, such as modular arithmetic, to identify and analyze the intersections between infinite primes sets. These techniques involve rigorous proofs and calculations to accurately determine the patterns and relationships between prime numbers.

What are some potential applications of understanding intersections of infinite primes sets?

Understanding intersections of infinite primes sets can have practical applications in fields such as cryptography, where prime numbers are used to secure data and information. It can also aid in the development of more efficient algorithms for prime factorization and prime number testing.

Are there any real-world examples of intersections of infinite primes sets?

Yes, there are many real-world examples of intersections of infinite primes sets. One famous example is the RSA encryption algorithm, which relies on the intersection of two sets of prime numbers to create a secure key for encrypted communication.

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