nice post and an interesting article. I've always been fascinated by primes but I lack the technical knowledge on the subject.. I try to understand the math but it's too complicated for an electronics engineer like me :D .. I had this silly question, forgive my naivety - is there any trend to the spacing between adjacent primes ? Or is it purely random? Like, if I plot the spacing between adjacent primes up to some number 'n' where n = very large, then will the plot be sheer randomness? Also, another question I always had was:- is there an upper limit to the spacing between adjacent primes? Has someone proved or disproved that such an upper limit exists?
@Digbijoy, thank you. As per current knowledge, we don't really know of a pattern for spacing between adjacent primes. But, we also can't say that the spacing is purely random. For example, a recent result of Kannan Soundararajan and Robert Lemke Oliver shows that a prime ending in a certain digit (say 1) is less likely to be followed by a prime ending in the same digit than in the other possible digits (3, 7, 9). Though it is not directly a statement about gaps between adjacent primes, it does say something about their location with respect to each other not being random. Link: https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/
Regarding your second question, consider the following string of numbers n! + 2, n! + 3, all the way up to n! + n. All of these are consecutive composite numbers. One can do this for any n. Thus, there is no upper limit to the spacing between adjacent primes.
The conjecture of Erdos (which I did not describe much in the article and which was proved by James Maynard and Terry Tao) suggests that as X becomes larger and larger, the size of the largest gap between adjacent primes up to X is greater than a function of the form
f(X) log X,
where f(X) itself increases rapidly with X. You can check out Terry Tao's blog for the precise function :-) https://terrytao.wordpress.com/2014/12/16/long-gaps-between-primes/
Wow ... very nicely explained reply, Kaneenika :) I appreciate the effort. I think I am getting a few of the fundas here and I looked up Wikipedia as well. Seems like there's uncountable conjectures and hypothesis on primes and things pertaining to primes such as spacing, density, twin primes, etc. etc. And even Golbach's conjecture isn't proved yet ! The derivations and analysis, mathematically, are so esoteric and complicated that without a PhD in pure maths, no one can grasp anything at all :D hehehe... I wonder if nature followed such a complicated route in prime distribution
4 comments:
Lovely! And an interesting training plan ;-)
nice post and an interesting article.
I've always been fascinated by primes but I lack the technical knowledge on the subject.. I try to understand the math but it's too complicated for an electronics engineer like me :D .. I had this silly question, forgive my naivety - is there any trend to the spacing between adjacent primes ? Or is it purely random? Like, if I plot the spacing between adjacent primes up to some number 'n' where n = very large, then will the plot be sheer randomness?
Also, another question I always had was:- is there an upper limit to the spacing between adjacent primes? Has someone proved or disproved that such an upper limit exists?
@Digbijoy, thank you. As per current knowledge, we don't really know of a pattern for spacing between adjacent primes. But, we also can't say that the spacing is purely random. For example, a recent result of Kannan Soundararajan and Robert Lemke Oliver shows that a prime ending in a certain digit (say 1) is less likely to be followed by a prime ending in the same digit than in the other possible digits (3, 7, 9). Though it is not directly a statement about gaps between adjacent primes, it does say something about their location with respect to each other not being random. Link: https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/
Regarding your second question, consider the following string of numbers n! + 2, n! + 3, all the way up to n! + n. All of these are consecutive composite numbers. One can do this for any n. Thus, there is no upper limit to the spacing between adjacent primes.
The conjecture of Erdos (which I did not describe much in the article and which was proved by James Maynard and Terry Tao) suggests that as X becomes larger and larger, the size of the largest gap between adjacent primes up to X is greater than a function of the form
f(X) log X,
where f(X) itself increases rapidly with X. You can check out Terry Tao's blog for the precise function :-)
https://terrytao.wordpress.com/2014/12/16/long-gaps-between-primes/
Wow ... very nicely explained reply, Kaneenika :)
I appreciate the effort. I think I am getting a few of the fundas here and I looked up Wikipedia as well. Seems like there's uncountable conjectures and hypothesis on primes and things pertaining to primes such as spacing, density, twin primes, etc. etc.
And even Golbach's conjecture isn't proved yet !
The derivations and analysis, mathematically, are so esoteric and complicated that without a PhD in pure maths, no one can grasp anything at all :D hehehe... I wonder if nature followed such a complicated route in prime distribution
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