Does having prime neighbors make you more composite?

147 点作者 another超过 3 年前

17 条评论

This is a lovely friday morning diversion. I won't comment on the mathematical content, since the article did such an enjoyable job, but I do want to call out this slightly tangential quote:> Could it be that I’m the first person ever to notice the curious properties of twin tweens? No. I am past the age of entertaining such droll thoughts, even transiently. If I have not found any references, it’s doubtless because I’m not looking in the right places.Ever since I read Neal Stephenson's Anathem, this idea has stuck with me. Especially in regards to questions about who deserves the glory for creating/popularizing things first. Vanishingly few ideas are new, and that's okay. Journey, not destination, yada yada... What a fun write-up.

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rjmunro超过 3 年前

I think of it like this. If a number n has a factor f, f cannot be a factor of n+1 or n-1 (unless f is 1, but we can ignore that for primeness). The next numbers to have f as a factor are n+f or n-f.If a number n has loads of factors, all of those factors are excluded from n+1 and n-1, so there are not many numbers left to be factors of them and they are likely to be twin primes.

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AnotherGoodName超过 3 年前

Oh the site is going slow but it did load.>Another cross reference took me off to sequence A002822, labeled “Numbers m such that 6m−1, 6m+1 are twin primes"Don't focus on 6m+/-1 alone and you'll get it. Let me explainAll prime numbers above 2 are in the form of 2m + 1. ie. They are odd. So we have a formula that rules out half of numbers from being possible prime.Now we can do a similar thing and create a formula to rule out factors of 2 and 3. Factors of 2 and 3 repeat every 6 numbers (the multiple of 2 and 3). eg. 6m + 0, 6m + 2 or 6m + 4 is divisible by 2. 6m + 0 or 6m + 3 is divisible by 3.So all prime numbers above 3 are in the form of 6m + 1 or 6m + 5. We can write the 6m + 5 as 6m - 1 if we want. Only 2 out of 6 numbers can be prime above 3 and the numbers either side of these will have either a factor of 2 or a factor of 6. (Btw never simplify the 'only 2 out of 6 numbers can be prime' as there are windows (twin primes for example) where there's more than 1 our of 3 numbers that are prime. It's only true that there's never more than 2 out of 6 sequential numbers prime above 6. If you come up with a prime number theory that sets a maximum of the frequency of primes you need to consider the window size rather than a straight fraction or you'll end up with errors).Now we can do the same as above for 5. 30 is where factors of 2,3 and 5 align. All prime numbers above 30 are in the form of 30m + 1, 7, 11, 13, 17, 19, 23, 29. The rest are factors or 2,3 or 5. Only 8 out of 30 numbers above 5 can possibly be prime. All those primes are surrounded by a factor of 6 and 2 (since 6m +1/-1 is a subset of this) or in the +1/+29 case surrounded by a factor of 2 and a factor of 30. For the twin prime case 1/3 of prime numbers are next to multiples of 2,3,5 and 2/3 are next to multiples of 2,3.You could do the same with 2x3x5x7 = 210. And again come up with a formula for the maximum frequency of primes (48 out of 210 numbers above 7 can possible by prime) and also a formula for where primes can occur. In this case you'd see for the twin prime case 1/21 are next to 2x3x5x7, 6/21 next to 2x3x5, 14/21 next to 2x3.And we can keep doing this again and again. Each time we'll see the frequency for primes decrease and we'll see a new possibility for an even more composite number next to a prime opening up.Now we can see clearly that primes are always next to at least one composite number. As the numbers get larger and larger there's there's more possibility for the number next to a prime being more composite.

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howmayiannoyyou超过 3 年前

Off topic, but: Reading the post title made me wonder: Does having neighbors with Amazon Prime Memberships result in getting your purchases faster even if you don't have a membership?

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roberto超过 3 年前

This is how I understand it:For every 3 consecutive numbers, one of them is a multiple of 3. If a number has 2 prime neighbors there's 100% chance it's divisible by 3. Without prime neighbors, only 33%.For 3 consecutive numbers there's a 75% chance one of them is a multiple of 4. The number with 2 prime neighbors has then a 75% chance of being divisible by 4. A number without prime neighbors has only 25%.For 5, it's 60% vs 20%.So on average we expect the numbers with prime neighbors to be more composite.

sweezyjeezy超过 3 年前

Here's a relevant stackexchange post <a href="https://math.stackexchange.com/questions/3490592/what-is-notable-about-the-composite-numbers-between-twin-primes" rel="nofollow">https://math.stackexchange.com/questions/3490592/what-is-not...</a>TLDR - given some very reasonable (and wildly unproven) heuristics about primes, a large composite number between twin prime is expected to have approximately 2.180950 times as many divisors as usual.

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ouid超过 3 年前

odds are, if you have prime neighbors, you are even.

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phkahler超过 3 年前

I would like to point out that if we loosen some definitions a bit to include negative numbers, we could claim 1 and -1 as some form of "primes" and 0 is an integer multiple of an infinite number of primes (all of them).There is something fundamental about zero that I've been trying to figure out how to state clearly, but haven't been able to put in words just yet. It is closely related to the above.

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dghughes超过 3 年前

>Babylonian accountants and land surveyors did their arithmetic in base 60, presumably because sexagesimal numbers help with wrangling fractions.I thought the agreed theory is its due to Babylonians counting the sections of each finger? The thumb is the pointer and it's used to touch each of the 12 sections five times over one loop per finger (thumb sections are not included). It's also why clocks and time are base 12.

Jap2-0超过 3 年前

Just looking through the comments: a lot of people seem to be (knowingly or not) describing the Sieve of Eratosthenes.[0][0] <a href="https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes" rel="nofollow">https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes</a>

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edw519超过 3 年前

A long time ago, I gave up the pursuit of Elementary Number Theory for a career in programming.Thanks OP for reminding me there's more than one way for a nerd to happy dance. Great post!

mark-r超过 3 年前

I wonder if reversing the statement is useful? Are the neighbors of a highly composite number more likely to be prime?

vgeek超过 3 年前

So should having prime neighbors be a factor when looking at multiples of properties?

curtisf超过 3 年前

Here's one way to bolster this mathematically just a bit.If you have a composite number C, you can use C to guarantee factors in larger numbers. If k is a factor of C, Cn + k also has a factor of k.So, for a "highly-composite" C, you rule out many prime numbers. An easy way to generate these "highly composite" numbers is to multiple the first few primes. For example,C = 2 * 3 = 6.* 6n + 0 has factors of 2, 3, 6* 6n + 1 may be prime (e.g. 7)* 6n + 2 has factors of 2* 6n + 3 has factors of 3* 6n + 4 has factors of 2* 6n + 5 may be prime (e.g., 17)I'll call the possibly-prime numbers C-primes. So 6n+1 and 6n+5 are 6-primes. Similarly, we have 6n+0 is a 6-tween, and 6n+2 and 6n+4 are 6-nexts (next to exactly one 6-prime), and 6n+3 is a 6-non (not adjacent to any 6-primes).So, the 6-tweens have at least 2 factors, the 6-nons have at least 1 factor, and 6-nexts have at least 1 factor.The number of proper divisors that a number can have is actually bounded fairly tightly [1]. For example, number numbers below 24 have more than 4 divisors; that means the 6-tween behavior predicts _three quarters_ of the divisors a number can have until 24.[1]: <a href="https://terrytao.wordpress.com/2008/09/23/the-divisor-bound/" rel="nofollow">https://terrytao.wordpress.com/2008/09/23/the-divisor-bound/</a>You can use a larger C to get information for longer. For example, C = 2 * 3 * 5 = 30 gives* 30n + 0 has divisors 2, 3, 5, 6, 10, 15, 30 -- tween* 30n + 1 may be prime* 30n + 2 has divisors 2 -- next* 30n + 3 has divisors 3* 30n + 4 has divisors 2* 30n + 5 has divisors 5* 30n + 6 has divisors 2, 3, 6 -- next* 30n + 7 may be prime* 30n + 8 has divisors 2 -- next* 30n + 9 has divisors 3* 30n + 10 has divisors 2, 5, 10 -- next* 30n + 11 may be prime* 30n + 12 has divisors 2, 3, 6 -- tween* 30n + 13 may be prime* 30n + 14 has divisors 2 -- next* 30n + 15 has divisors 3, 5, 15* 30n + 16 has divisors 2 -- next* 30n + 17 may be prime* 30n + 18 has divisors 2, 3, 6 -- tween* 30n + 19 may be prime* 30n + 20 has divisors 2, 5, 10 -- next* 30n + 21 has divisors 3* 30n + 22 has divisors 2 -- next* 30n + 23 may be prime* 30n + 24 has divisors 2, 3, 6 -- next* 30n + 25 has divisors 5* 30n + 26 has divisors 2* 30n + 27 has divisors 3* 30n + 28 has divisors 2 -- next* 30n + 29 may be primeSo 30-tweens have (7, 3, 3) divisors; 30-nexts have (1, 3, 3, 1, 1, 3, 3) divisors; 30-nons have (1, 1, 1, 1, 3, 1, 1, 1, 1) divisors.The first number to have more than 10 divisors is 120 (with 14), so this predicts 7 of the possible 13 divisors for numbers 30 to 120, all in tween numbers.This makes the problem somewhat more concrete: why do the numbers that have many common factors with C cluster near the numbers with no common factors of C? But since we can at least observe it for different concrete C, this still does predict something about "all" the primes (though the effect of a particular C fades as the number of possible divisors eventually increases >> num. divisors of C; I posit that you can always generate a larger C to extend the pattern, but I don't know number theory to show it)

kerblang超过 3 年前

I was actually expecting this was about Amazon collecting data on the neighbors of Prime customers by forming some composite view of not-Prime vs. Prime traffic. My knee seems to be trained to jerk in a persistent direction...

KingOfCoders超过 3 年前

Prime neighbours? 3,5,7?

intrasight超过 3 年前

I'm in #4. My neighbors are prime across the street (#3 and #5) and non-prime on my side (#2 and #6)

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