Is it possible to locate prime numbers through addition only

In summary, the conversation discusses various methods for finding and generating prime numbers, including the Eratosthenes method, prime-generating polynomials, and computer algorithms. The participants also mention the difficulty of finding a pattern or algorithm that works for all primes, and the existence of a web tool for factorizing large prime numbers.
  • #1
Carl A Bohn
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0
I was reading an old thread about multiplying successive prime numbers adding 1 to obtain another prime number.
I have worked with prime numbers for several years now and have developed what I best call a bi-linear advancement. It is an open-ended sieve of Eratosthenes. After many, many hours across the years, I have finally developed a piece of code that locates prime numbers. I built it on a small laptop in QB6, and so far it is locating primes out in the 4.9 billion range. (Time elapsed: 18 hours.)
Actually, I am locating all the prime-subs. From there, I am able to extract all the intervening prime numbers.
The one key issue I had to deal with is the prime numbers 2,3, and 5. I found a way to extract those multiples from the process, then discovered a rather unique way of grouping prime numbers. The rest of it is just plain math. But, I took it a step further a developed a system of linear tables that reduces the whole process down to a lookup table. I also have a very unique way to store prime numbers in a very condensed form.
 
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  • #2
Folks have been trying to come up with a magic prime number generator for centuries and all have failed.

The Eratosthenes method is perhaps the only one. It was used to prove that primes are infinite ie that having found all primes to a certain point it’s possible to make a composite number as the product of all these primes plus one to be a new prime.

If you use it as a generator it misses primes.

We start with 2 as the first prime, so then 2+1=3 the next prime.

Next we say 2*3+1 = 7 which is prime but we have skipped 5 and the process breaks as a generator of primes.

I’m sure we could somehow fix the algorithm but then some higher prime will be missed and so it goes. The primes are just impossible to work with they appear to have no known pattern for generation for all primes although we might find an algorithm that works some of the time.
 
  • #3
These sieves work well for small numbers, but finding primes for small numbers is a trivial task for computers anyway. Computers can factorize 40-digit primes numbers in less than a second, and check 100-digit numbers for primality in the same time.

Here is a web tool if you want to try it yourself. A random 47-digit number which is hard to factorize is 28701392776125735524335735103358699296374619111.
 
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  • #4
Sorry about off-topic, but yesterday I was thinking about prime numbers and wondered whether for any finite set of integers ##A = \{a_1 ,a_2 , \dots ,a_n \}## it is possible to find a base number ##b \in \mathbb{R}## such that ##\log_b a## is rational for all ##a \in A##. This is probably not possible, but if it were, then the logarithm of any integer that is a product of integers belonging in ##A## would also be rational.
 
  • #5
mfb said:
Computers can factorize 40-digit primes in less than a second
Humans can factorize primes in O(1).
 
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  • #6
mfb said:
Here is a web tool if you want to try it yourself. A random 47-digit number which is hard to factorize is 28701392776125735524335735103358699296374619111.
Nice program @mfb ! I tested it on :
1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567891

Took about 1:45 hours on my computer to factorize. Now I am testing it on the latest version of Mathematica. See which one is faster.

EDIT--- Mathematica rocks! 30 mins with FactorInteger.
 
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  • #7
@Carl A Bohn Do not waste your time. Regarding one of your previous threads (now closed) a prime-generating polynomial has been found by James Jones, Daihachiro Sato, Hideo Wada and Douglas Wiens in 1977. Yes, it finds all primes!
 
  • #8

1. Can prime numbers be located through addition only?

Yes, it is possible to locate prime numbers through addition only. This method is known as the Sieve of Eratosthenes, which is a mathematical algorithm used to generate prime numbers.

2. How does the Sieve of Eratosthenes work?

The Sieve of Eratosthenes works by creating a list of numbers from 2 to a given maximum number. Then, starting with the first prime number (2), all multiples of 2 are eliminated from the list. This process is repeated with the next remaining number (3) and its multiples, and so on until there are no more numbers left in the list. The remaining numbers in the list are the prime numbers.

3. What is the benefit of using addition only to locate prime numbers?

The benefit of using addition only to locate prime numbers is that it is a simple and efficient method, especially for smaller numbers. It also allows for a visual representation of prime numbers and their patterns.

4. Are there any limitations to locating prime numbers through addition only?

Yes, there are limitations to this method. It becomes less efficient and more time-consuming for larger numbers. It also does not work for finding extremely large prime numbers, as it would require a significantly larger list of numbers and a longer elimination process.

5. Are there other methods besides the Sieve of Eratosthenes to locate prime numbers through addition only?

Yes, there are other methods such as the Sieve of Sundaram and the Sieve of Atkin. These methods use similar principles to the Sieve of Eratosthenes but may be more efficient for specific types of numbers.

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