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Post a number with an interesting property

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Rudin
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18 Oct 2015, 5:28 pm

16:

16=2^4=4^2

or

There exists two integers m and n such that,

16=m^n=n^m

There are no other numbers with that property.


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naturalplastic
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19 Oct 2015, 6:02 am

Not so!

27 also has that property!

If if you take "three cubed", and flip it around its..."three cubed". And three cubed equals...three cubed!



Rudin
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19 Oct 2015, 6:13 am

naturalplastic wrote:
Not so!

27 also has that property!

If if you take "three cubed", and flip it around its..."three cubed". And three cubed equals...three cubed!


Ah. I forgot to mention, m cannot be equal to n.


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"God may not play dice with the universe, but something strange is going on with prime numbers."

-Paul Erdos

"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

-Bruce Schneider


LonelyJar
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19 Oct 2015, 6:17 am

Sheldon Cooper: "The best number is 73. Why? 73 is the 21st prime number. Its mirror, 37, is the 12th and its mirror, 21, is the product of multiplying 7 and 3... and in binary 73 is a palindrome, 1001001, which backwards is 1001001."



Rudin
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19 Oct 2015, 6:27 am

Why not just say,

"73 is the 21st prime and 21 is a product of multiplying 3 and 7 etc."?

There are also far cooler numbers than 73.


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"God may not play dice with the universe, but something strange is going on with prime numbers."

-Paul Erdos

"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

-Bruce Schneider


Rudin
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19 Oct 2015, 6:31 am

Sophomore's Dream is an awesome number, one of my favourites.

Image


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"God may not play dice with the universe, but something strange is going on with prime numbers."

-Paul Erdos

"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

-Bruce Schneider


Last edited by Rudin on 19 Oct 2015, 6:32 am, edited 1 time in total.

Rudin
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19 Oct 2015, 6:31 am

Sophomore's Dream is the first equality.


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"God may not play dice with the universe, but something strange is going on with prime numbers."

-Paul Erdos

"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

-Bruce Schneider


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19 Oct 2015, 6:54 am

Rudin wrote:
Why not just say,

"73 is the 21st prime and 21 is a product of multiplying 3 and 7 etc."?


Because there is that added feature- that if you flip 73 to make it 37- you have the 12th prime number, and 12 is the mirror of 21.

So 73 is pretty cool.



slave
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19 Oct 2015, 3:16 pm

42

:wink: :nerdy:



Rudin
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19 Oct 2015, 5:29 pm

slave wrote:
42

:wink: :nerdy:


Ah, a Hitchhiker's Guide to the Galaxy reference.


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"God may not play dice with the universe, but something strange is going on with prime numbers."

-Paul Erdos

"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

-Bruce Schneider


GoofyGreatDane
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20 Oct 2015, 2:17 pm

Aleph-null : the cardinality of the set of natural numbers.
A cardinality is like the "number of elements in the set". For any set of finite cardinality, adding an element generates a set with a larger cardinality. {} has cardinality zero, while {a} has cardinality of one.

Aleph-null is the cardinality of the set of positive integers. As an infinite cardinal , it has some interesting properties.
One interesting property is that aleph-null + 1 =aleph-null , adding one element to an infinite set, or even a countably infinite number of elements to the set, does not change the cardinality of the set. So the set of even numbers , the set of all rational numbers, and the set of integers are of the same size. One set that is guartanteed to be larger than a given infinite set is the set of all subsets of the set.

Another interesting property of infinite cardinals is the continuum hypothesis. The generalized continuum hypothesis states that there is no set with cardinality between the that of an infinite set and the set of all subsets of of that set. It would imply that there is no set with cardinality between that of the natural numbers and that of the real numbers. This would make sense but has never been proven. And more interestingly, not only has this never been proven- but its also been shown to be independent of ZFC axiomatic set theory. This means that both the hypothesis and its negation are equally "valid " in ZFC- there is no way to construct a set in ZFC that you can prove has an intermediate cardinality, but you can't prove that there is no set with such an intermediate cardinality either.



Rudin
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20 Oct 2015, 6:57 pm

GoofyGreatDane wrote:
Aleph-null : the cardinality of the set of natural numbers.
A cardinality is like the "number of elements in the set". For any set of finite cardinality, adding an element generates a set with a larger cardinality. {} has cardinality zero, while {a} has cardinality of one.

Aleph-null is the cardinality of the set of positive integers. As an infinite cardinal , it has some interesting properties.
One interesting property is that aleph-null + 1 =aleph-null , adding one element to an infinite set, or even a countably infinite number of elements to the set, does not change the cardinality of the set. So the set of even numbers , the set of all rational numbers, and the set of integers are of the same size. One set that is guartanteed to be larger than a given infinite set is the set of all subsets of the set.

Another interesting property of infinite cardinals is the continuum hypothesis. The generalized continuum hypothesis states that there is no set with cardinality between the that of an infinite set and the set of all subsets of of that set. It would imply that there is no set with cardinality between that of the natural numbers and that of the real numbers. This would make sense but has never been proven. And more interestingly, not only has this never been proven- but its also been shown to be independent of ZFC axiomatic set theory. This means that both the hypothesis and its negation are equally "valid " in ZFC- there is no way to construct a set in ZFC that you can prove has an intermediate cardinality, but you can't prove that there is no set with such an intermediate cardinality either.


The set of algebraic reals is surprisingly equivalent to the integers, but much, much more dense.


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"God may not play dice with the universe, but something strange is going on with prime numbers."

-Paul Erdos

"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

-Bruce Schneider


Feyokien
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20 Oct 2015, 7:14 pm

slave wrote:
42

:wink: :nerdy:


+1



naturalplastic
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20 Oct 2015, 7:48 pm

Rudin wrote:
1729 is also the product of three distinct primes (sphenic number).

It is also a Carmichael number. Meaning it is an exception to Fermat's primality test. Watch Numberphile for an explanation.


It was also the year that Antonio Vivaldi composed 'The Four Seasons'.



naturalplastic
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21 Oct 2015, 2:12 pm

1337.

Its become slang for "elite" because if you turn it upside down its spells "LEET".

And 1337 (im pretty sure, if I am not mistaken) is also a prime number. And being prime makes it a "LEET" number to mathematicians.



slave
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21 Oct 2015, 4:53 pm

57 is 111 in base 7.
259 is 1111 in base 6.
400 is 1111 in base 7.
781 is 11111 in base 5.
255 is 11111111 in base 2.
511 is 111111111 in base 2.