This is one of my favourite mysteries.
https://en.wikipedia.org/wiki/Goldbach%27s_conjecture
How is it that all even number are the sum of 2 primes and odd number the sum of 3 primes?
Why do prime numbers make these spirals? (2019)
51–60 of 141 posts
Re: Why do prime numbers make these spirals? (2019)
#52One of the thrills of studying the primes is the discovery that all primes greater than 3 are of the form 6k+1 or 6k-1. And for primes greater than 2, all primes are of the form 4k+1 or 4k-1. It is something that is commonly rediscovered by students, and that new independent finding was quite exciting for me. The reasoning, which is in the article here, is that you can make any whole number you wish if the number is…
I have discovered the same thing while practicing prime sieve algorithms back in the day. Such properties of primes are quite useful for optimizing both speed and memory of sieve algorithms.
More generally, if you take the first n primes p_1, ..., p_n and define P=p_1*...*p_n, then all primes bigger than P be in the form of P*k +- a, such that a and GCD(P, a) = 1. In case of n=2 (P=2*3=6), there is this nice property that the only such a are 1 and 5 (which are equivalent), but in principle, the same can be done for any n. It's just that the set of all a has the size equal to Euler's totient function of P, which grows pretty fast as n increases.
For example, if n = 3, then P = 2*3*5 = 30, so all prime numbers bigger than 30 have to be in the form of 30k +- 1, 30k +- 7, 30k +- 11, 30k +- 13, 30k +- 17, 30k +- 19, 30k +- 23 or 30k +- 29 (notice that half of these are equivalent to the other half and can be ommitted). It is interesting that in this particular case, all a are either 1 or a prime less than P: I don't think that property holds for all n, though.
Re: Why do prime numbers make these spirals? (2019)
#53One of the thrills of studying the primes is the discovery that all primes greater than 3 are of the form 6k+1 or 6k-1. And for primes greater than 2, all primes are of the form 4k+1 or 4k-1. It is something that is commonly rediscovered by students, and that new independent finding was quite exciting for me. The reasoning, which is in the article here, is that you can make any whole number you wish if the number is…
You can also approximate pi as 22/7, which explains some of the other patterns that appear farther out.
Re: Why do prime numbers make these spirals? (2019)
#54This is one of my favourite mysteries. https://en.wikipedia.org/wiki/Goldbach%27s_conjecture How is it that all even number are the sum of 2 primes and odd number the sum of 3 primes?
We don't know whether it's true.
Re: Why do prime numbers make these spirals? (2019)
#55My browser did not enjoy that 40MB web page.
Re: Why do prime numbers make these spirals? (2019)
#56One of the thrills of studying the primes is the discovery that all primes greater than 3 are of the form 6k+1 or 6k-1. And for primes greater than 2, all primes are of the form 4k+1 or 4k-1. It is something that is commonly rediscovered by students, and that new independent finding was quite exciting for me. The reasoning, which is in the article here, is that you can make any whole number you wish if the number is…
Its the dimensionality? 2nd dimension, 3rd dimension and there permutation shadows? So by the mathematical principal, there is only one interesting permutation, and thats the "natural" one within the first dimension, aka 1 and 2s.
Re: Why do prime numbers make these spirals? (2019)
#57This is one of my favourite mysteries. https://en.wikipedia.org/wiki/Goldbach%27s_conjecture How is it that all even number are the sum of 2 primes and odd number the sum of 3 primes?
We don't know whether it's true.
Well, then it's a great mystery of up to 10^18: "T. Oliveira e Silva ran a distributed computer search that has verified the conjecture for n ≤ 4 × 10^18"
Re: Why do prime numbers make these spirals? (2019)
#58[dead]
Re: Why do prime numbers make these spirals? (2019)
#59This is one of my favourite mysteries. https://en.wikipedia.org/wiki/Goldbach%27s_conjecture How is it that all even number are the sum of 2 primes and odd number the sum of 3 primes?
The odd number sums follow from the fact about the even numbers.
Re: Why do prime numbers make these spirals? (2019)
#60One of the thrills of studying the primes is the discovery that all primes greater than 3 are of the form 6k+1 or 6k-1. And for primes greater than 2, all primes are of the form 4k+1 or 4k-1. It is something that is commonly rediscovered by students, and that new independent finding was quite exciting for me. The reasoning, which is in the article here, is that you can make any whole number you wish if the number is…
You can also approximate pi as 22/7, which explains some of the other patterns that appear farther out.
> You can also approximate pi as 22/7
Except in Indiana, where the legal value of pi is 3.2 by mandate.