Large Gaps between Primes - Numberphile
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My best explanation would be that this gap would come infinitely far out, so there would already be an infinite number of primes, but I'm sure a lot of you have smarter things to say! :D
Just for the record: I know I'm treating infinity in a messy way, and that is probably not a very wise decision.
Hope you can help me: )
Date: 2022-04-08
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Comments and reviews: 9
Raymond
The MATRIX and gaps. At 2-3-5 = 30n the primes are 31 and 37 gap of 6. At 210n the primes are 211 and 223 gap of 12. At 2310n the primes are 2311 and 2333 gap 22. At 30030n the prime is 30047 gap 17. At 510510n the prime is 510529 gap 19. At 9699690n the prime 9699713 gap 23. Let move to 13082761331670030n the prime is 13082761331670077 gap is 47. The MATRIX shows a steady increase in the gaps as prime numbers are multiplied together. Using all primes up to 281 creates a 116 digit number with a gap of 310. The -n- notes that each factorial# is repeatable infinite times. This MATRIX is as infinite as prime numbers. NOTE: At 100 million digits the gap exceeds 230 million.
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The MATRIX and gaps. At 2-3-5 = 30n the primes are 31 and 37 gap of 6. At 210n the primes are 211 and 223 gap of 12. At 2310n the primes are 2311 and 2333 gap 22. At 30030n the prime is 30047 gap 17. At 510510n the prime is 510529 gap 19. At 9699690n the prime 9699713 gap 23. Let move to 13082761331670030n the prime is 13082761331670077 gap is 47. The MATRIX shows a steady increase in the gaps as prime numbers are multiplied together. Using all primes up to 281 creates a 116 digit number with a gap of 310. The -n- notes that each factorial# is repeatable infinite times. This MATRIX is as infinite as prime numbers. NOTE: At 100 million digits the gap exceeds 230 million.
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Andy
ok, there is a question bothering me:
Accepting we are able to proove, that there is an albitrarily large (countably infinite) gap in between at least two primes, does that mean the following for us:
concerning practical matters of reality: will we find a prime, that is -the largest prime-, we will ever be capable to find, as the next one is going to be infinitely far away and therefore uncalculatably -far away-? Or is it just one of these -infinity--paradoxes (what I expect the unsatisfying answer to be, where we will find, that it's just a question of computational power or a question of definiton of infinity?
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ok, there is a question bothering me:
Accepting we are able to proove, that there is an albitrarily large (countably infinite) gap in between at least two primes, does that mean the following for us:
concerning practical matters of reality: will we find a prime, that is -the largest prime-, we will ever be capable to find, as the next one is going to be infinitely far away and therefore uncalculatably -far away-? Or is it just one of these -infinity--paradoxes (what I expect the unsatisfying answer to be, where we will find, that it's just a question of computational power or a question of definiton of infinity?
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Justin
Someone point out the flaw in my thinking here. so you could have an infinitely large gap between prime numbers. But if you have an infinitely gap between prime numbers, you never get to the next prime number. but we know that there are infinitely many primes with a gap of at most 6. So one the one hand, I can do n! +n and have n approach infinity and show that there is an infinite gap between primes at some point, but we've also shown that there are infinitely many primes with a gap of 6. Can someone help me wrap my head around this using grade school language? Thanks!
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Someone point out the flaw in my thinking here. so you could have an infinitely large gap between prime numbers. But if you have an infinitely gap between prime numbers, you never get to the next prime number. but we know that there are infinitely many primes with a gap of at most 6. So one the one hand, I can do n! +n and have n approach infinity and show that there is an infinite gap between primes at some point, but we've also shown that there are infinitely many primes with a gap of 6. Can someone help me wrap my head around this using grade school language? Thanks!
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Fred
Okay, I was curious, so I just went on Desmos to take look at that function.
. yeah.
So, that's about gaps between prime numbers -if x is large. - How large? Well, to give you a sense, that function is not even DEFINED for any x less than 10 billion. And THEN it needs to climb all the way up from negative infinity! I don't even know when it reaches ZERO much less any reasonable number for prime gaps, because Desmos actually gives up all the way at about 10-300 - STILL several hundred below zero!
I mean, please.
That's just unreasonable.
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Okay, I was curious, so I just went on Desmos to take look at that function.
. yeah.
So, that's about gaps between prime numbers -if x is large. - How large? Well, to give you a sense, that function is not even DEFINED for any x less than 10 billion. And THEN it needs to climb all the way up from negative infinity! I don't even know when it reaches ZERO much less any reasonable number for prime gaps, because Desmos actually gives up all the way at about 10-300 - STILL several hundred below zero!
I mean, please.
That's just unreasonable.
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Robert
The number 2 should not be a prime. It shares no properties of the primes except the indivisibility property and this is by nature of its low value. It is an even number and it is also a power of 2. Its occurrence has nothing to do with the reason that primes occur. If one wants to be pedantic 3 should also not be a prime. It does not have the ultimate fundamental property that makes a prime a prime. Yes it is indivisible but this is not the quintessential property of the primes.
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The number 2 should not be a prime. It shares no properties of the primes except the indivisibility property and this is by nature of its low value. It is an even number and it is also a power of 2. Its occurrence has nothing to do with the reason that primes occur. If one wants to be pedantic 3 should also not be a prime. It does not have the ultimate fundamental property that makes a prime a prime. Yes it is indivisible but this is not the quintessential property of the primes.
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Metalkatt
Honest question: Why is it important? I get that there are a lot of people who are completely in love with prime numbers, but does knowing this help us do anything else? I'm not against knowledge for its own sake, nor against the -just because it's cool- factor or anything. Don't think I'm getting down on it. I'm just wondering if it helps us in any appliable (not applicable; I chose the 'word' deliberately) way.
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Honest question: Why is it important? I get that there are a lot of people who are completely in love with prime numbers, but does knowing this help us do anything else? I'm not against knowledge for its own sake, nor against the -just because it's cool- factor or anything. Don't think I'm getting down on it. I'm just wondering if it helps us in any appliable (not applicable; I chose the 'word' deliberately) way.
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TimGChannel
Has anybody else found this prime sieve? For a! +b=c, where b is a prime larger than a, what is the smallest b has to be in order for c to be prime? So far, I'm finding surprisingly small bs. The first twenty a, b are 1, 2; 2, 3; 3, 5; 4, 5; 5, 7; 6, 7; 7, 11; 8, 23; 9, 17; 10, 11; 11, 17; 12, 29; 13, 67; 14, 19; 15, 43; 16, 23; 17, 31; 18, 37; 19, 89; 20, 29. Is this anything?
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Has anybody else found this prime sieve? For a! +b=c, where b is a prime larger than a, what is the smallest b has to be in order for c to be prime? So far, I'm finding surprisingly small bs. The first twenty a, b are 1, 2; 2, 3; 3, 5; 4, 5; 5, 7; 6, 7; 7, 11; 8, 23; 9, 17; 10, 11; 11, 17; 12, 29; 13, 67; 14, 19; 15, 43; 16, 23; 17, 31; 18, 37; 19, 89; 20, 29. Is this anything?
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Dee
idea for a piece of mathy art: you have your x axis be the increment you increase by and your y axis is the prime you start with, and at every point you color it based on a scale from 0 to the highest number of primes in a row included on the graph. i'd be very interested to see how it turned out, but i don't have the brainpower, patience, or resources to do it myself.
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idea for a piece of mathy art: you have your x axis be the increment you increase by and your y axis is the prime you start with, and at every point you color it based on a scale from 0 to the highest number of primes in a row included on the graph. i'd be very interested to see how it turned out, but i don't have the brainpower, patience, or resources to do it myself.
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Brady
Here's a simple exclusionary primes test that takes very little mathematical knowledge or computation: all primes greater than 10 will have one of four particular ones-place digits: 1, 3, 7, or 9. Any number greater than 10 that ends with any of the other six digits cannot be prime; either it is even (divisible by 2, or divisible by 5, or both.
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Here's a simple exclusionary primes test that takes very little mathematical knowledge or computation: all primes greater than 10 will have one of four particular ones-place digits: 1, 3, 7, or 9. Any number greater than 10 that ends with any of the other six digits cannot be prime; either it is even (divisible by 2, or divisible by 5, or both.
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