Six Sequences - Numberphile
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For instance, the Golden Ratio is an irrational number, and the continued fraction expansion for it is an endless series of 1's as the coefficients, meaning that the geometric mean of the coefficients would be 1 for any number of iterations, rather than converging to the Constant as the number of terms in the expansion approaches infinity.
Similarly, any irrational number constructed in the same way so that all the coefficients in the fraction expansion have the same value would have a geometric mean equal to that value instead of the the Constant.
Date: 2022-04-08
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Comments and reviews: 9
James
Wieferich Primes are hard to explain, but the best I can give it to you is by simply showing it, Wieferich Primes we know of, 1093, so p=1093, 2-(p-1) which is 2-(1092) can be divided by 1093, and come out with an integer, whereas if you tried say p=5, (2-4)/5 isn't an integer. Because you can rewrite the conjecture 2-(p-1) = 1, it needs to come out with an integer, to be a Wieferich Prime, hope that sort of helped with understanding it
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Wieferich Primes are hard to explain, but the best I can give it to you is by simply showing it, Wieferich Primes we know of, 1093, so p=1093, 2-(p-1) which is 2-(1092) can be divided by 1093, and come out with an integer, whereas if you tried say p=5, (2-4)/5 isn't an integer. Because you can rewrite the conjecture 2-(p-1) = 1, it needs to come out with an integer, to be a Wieferich Prime, hope that sort of helped with understanding it
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Kurtis
What about this sequence?
1, 11, 21, 1211, 111221, 312211, 13112221.
It starts with 1 and the next element -describes- the previous, hence the second number in the sequence is -11- meaning -one 1- (describing the previous number. The third number is then -21- meaning -two 1s-, and so on. I'm sure there's some very interesting math regarding this sequence.
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What about this sequence?
1, 11, 21, 1211, 111221, 312211, 13112221.
It starts with 1 and the next element -describes- the previous, hence the second number in the sequence is -11- meaning -one 1- (describing the previous number. The third number is then -21- meaning -two 1s-, and so on. I'm sure there's some very interesting math regarding this sequence.
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William
Its a nice technique that helps with concentration. We are trained to see brown as a constructive material so writing on it makes us think we are doing more than just writing on paper. The tactile sound and feel of the paper also helps with concentration and I honestly think it sounds nice and prefer it over just normal paper or a white board
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Its a nice technique that helps with concentration. We are trained to see brown as a constructive material so writing on it makes us think we are doing more than just writing on paper. The tactile sound and feel of the paper also helps with concentration and I honestly think it sounds nice and prefer it over just normal paper or a white board
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razzamattazz
If it began with 0 then I'm pretty sure it would have to just be an infinite sequence of zeros. 0, 0, 0, 0, 0, 0, etc. How many times does 1 appear in the sequence? Well, the 1st term is 0 so it has to appear 0 times. How many times does 2 appear in the sequence? Well, the 2nd term is 0 so it appears 0 times, and so on and so forth
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If it began with 0 then I'm pretty sure it would have to just be an infinite sequence of zeros. 0, 0, 0, 0, 0, 0, etc. How many times does 1 appear in the sequence? Well, the 1st term is 0 so it has to appear 0 times. How many times does 2 appear in the sequence? Well, the 2nd term is 0 so it appears 0 times, and so on and so forth
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Angela
Maybe I misunderstood something, but if the first one is just the sequence of digits in a real number, and the integers in the sequence aren't actually used as numbers, it's not really significant as an integer sequence. It's not s very integ sequence at all, let alone the integest.
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Maybe I misunderstood something, but if the first one is just the sequence of digits in a real number, and the integers in the sequence aren't actually used as numbers, it's not really significant as an integer sequence. It's not s very integ sequence at all, let alone the integest.
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Venkat
Infinite fraction is a decimal rotation of digits. As the fraction increase the decimals are insignificant and so reduce to k constant. Two most significant and other reduce fast. 3 is the closest. These kind of things are wave guides. Mostly used for encryption FM and AM.
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Infinite fraction is a decimal rotation of digits. As the fraction increase the decimals are insignificant and so reduce to k constant. Two most significant and other reduce fast. 3 is the closest. These kind of things are wave guides. Mostly used for encryption FM and AM.
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Marcus
Omygod golombs sequence is absolutely extraordinary. Just explaining how 1 could only occur once and obviously since 2 doesn't occur once because then it would be 1, which it can't be and, therefore MUST be 2 AND occur TWICE is incredible. Golomb's was by FAR the best
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Omygod golombs sequence is absolutely extraordinary. Just explaining how 1 could only occur once and obviously since 2 doesn't occur once because then it would be 1, which it can't be and, therefore MUST be 2 AND occur TWICE is incredible. Golomb's was by FAR the best
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SD
Levy's constant applies to itself and -almost all- numbers in the same sense that Khinchine's does. It's a related property of continued fractions. I nominate A087602 (its decimal expansion) and A086703 (its continued fraction expansion) as my favorites.
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Levy's constant applies to itself and -almost all- numbers in the same sense that Khinchine's does. It's a related property of continued fractions. I nominate A087602 (its decimal expansion) and A086703 (its continued fraction expansion) as my favorites.
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jamespfp
I called his favorite after he described its self-referential completeness. Ascribing divinity to it -- I tend to think of that as sentimentality, but it also gave me a chuckle. Nothing is as charming (at present) as completeness, eh?
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I called his favorite after he described its self-referential completeness. Ascribing divinity to it -- I tend to think of that as sentimentality, but it also gave me a chuckle. Nothing is as charming (at present) as completeness, eh?
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