 What is unusual about the # 1634? sonhouse Posers and Puzzles 16 Nov '08 21:00
1. 16 Nov '08 21:00
Regular base 10 number, nothing tricky there. 1634, what makes it unusual?
2. 16 Nov '08 21:02
Originally posted by sonhouse
Regular base 10 number, nothing tricky there. 1634, what makes it unusual?
It's a unique number, there is no other number with the same value.

On the other hand, every number are unique, so there is nothing more unique with this number than any other number, they are equally unique all of them.

Right?
3. 17 Nov '08 01:24
Originally posted by FabianFnas
It's a unique number, there is no other number with the same value.

On the other hand, every number are unique, so there is nothing more unique with this number than any other number, they are equally unique all of them.

Right?
Well, it has a certain property not many other numbers have. Got one correct reply by email already.
4. 17 Nov '08 04:07
The mathematical property of being exactly one more than 1633?

I did factor it, and 1634 = 2 * 19 * 43. Not sure if that sheds any light on the particular uniqueness though.
5. 17 Nov '08 04:28
1 + 1296 + 81 + 256 = 1634, which has 4 digits...
6. 17 Nov '08 04:54
http://en.wikipedia.org/wiki/1634

http://www.stetson.edu/~efriedma/numbers.html

http://mathworld.wolfram.com/NarcissisticNumber.html
7. 17 Nov '08 06:30
Statement (p): Every positive integer has an important property.

Anti-statement (-p): There is a number that has no important property.
The lowest number of all non-important number must be important being the lowest one. Then there is another non-important number. Apply this number to (-p).

Therefore the statement (p) holds.

Conclusion: Every number
8. 17 Nov '08 09:44
Conclusion: Every positive integer has an important property.

(Don't know why it was cut off...)
9. 17 Nov '08 13:19
Originally posted by coquette
http://en.wikipedia.org/wiki/1634

http://www.stetson.edu/~efriedma/numbers.html

http://mathworld.wolfram.com/NarcissisticNumber.html
Isn't it funny, in Wolfram's column, he devotes an awful lot of time to the numbers that are of no interest to mathemeticians🙂
10. 17 Nov '08 16:56
Originally posted by FabianFnas
Statement (p): Every positive integer has an important property.

Anti-statement (-p): There is a number that has no important property.
The lowest number of all non-important number must be important being the lowest one. Then there is another non-important number. Apply this number to (-p).

Therefore the statement (p) holds.

Conclusion: Every number
I've seen this in a few places now, and I still think it's great 🙂.
11. 18 Nov '08 19:15
Well, 1634 has the same property as 153. It's the sum of it's digits' powers of number of digits.

1634 = 1^4 + 6^4 + 3^4 + 4^4 = 1 + 1296 + 81 + 256
153 = 1^3 + 5^3 + 3^3 = 1 + 125 + 27

However, 153 has some other properties, for example, 153 = 1! + 2! + 3! + 4! + 5! = 1 + 2 + 3 + ... + 15 + 16 + 17. However, the later two properties wouldn't be so special, if they didn't refer to the same number together with the aforementioned property.
12. 19 Nov '08 21:111 edit
Originally posted by FabianFnas
Statement (p): Every positive integer has an important property.

Anti-statement (-p): There is a number that has no important property.
The lowest number of all non-important number must be important being the lowest one. Then there is another non-important number. Apply this number to (-p).

Therefore the statement (p) holds.

Conclusion: Every number
That sounds great, but does not fit.
The first "number with nothing special about" is great, but nobody
is interested in the 2nd one.

Who was the 2nd man on the moon?
Who flew as 2nd man across the atlantic ocean?
Who was the 2nd fastest mouse in all of mexico?

Nobody cares. The winner takes it all.
13. 20 Nov '08 07:21
Originally posted by afx
That sounds great, but does not fit.
The first "number with nothing special about" is great, but nobody
is interested in the 2nd one.

Who was the 2nd man on the moon?
Who flew as 2nd man across the atlantic ocean?
Who was the 2nd fastest mouse in all of mexico?

Nobody cares. The winner takes it all.
You missed the most important step in the proof.
14. 20 Nov '08 12:32
Originally posted by FabianFnas
Statement (p): Every positive integer has an important property.

Anti-statement (-p): There is a number that has no important property.
The lowest number of all non-important number must be important being the lowest one. Then there is another non-important number. Apply this number to (-p).

Therefore the statement (p) holds.

Conclusion: Every number
i do love this, but i was looking forward to contributing a great wikipedia article i read once about mathematical paradoxes and systems of reference... but i can't seem to find it! pretty sure it was in reading about Hilbert/Russel/Godel, etc. if anyone finds it it's a great link.

that being said, the main issue with this proof (i think) is not only in its self-reference - the assumption that the lowest of a group is thereby important - but also in it's CHANGE of system of reference. this is similar to the (in)famous proof that "all numbers are interesting" in that we seek to prove that all numerical identities have a property that is defined only in language and has no definable mathematical meaning... and then in order to prove that all numbers share that property, we apply the linguistic notion of "importance" to a numerical property of an element of a list of numbers. it reminds me of the grammatical error of comparisons in the sentence: "Mick Jagger's voice is the better than all the singers in America." it is intuitively understood by the reader, but is incorrect and logically false to compare a voice to a group of people (instead it should be voice -> voice, or person -> people). similarly, a numerically defined concept, such as "lowest," should not be related to a linguistically defined concept like "important."

i.e. it is inappropriately (though naturally) assumed that the smallest number in a list is also thereby important. this of course speaks to each reader as an intuitive idea.. but if you disregard that assumption the paradox breaks down.
15. 20 Nov '08 23:051 edit
Originally posted by TheMaster37
You missed the most important step in the proof.
No, I did not.
The induction step only holds, if the 2nd "number with nothing special" is "special", because the first NWNS is special.
But thats not the case, because nobody is interested in the 2nd one.
The induction step here assumes, that the 2nd one is as interesting
as the first one.
ps: I know, what a proof is, especially an induction proof, I am a mathematician.
Edit: The problem with this funny "proof" is the funny precondition. Because "being of interest" is being a thing of human "feeling", which is not fixed. So all the rest is a thing of good vibrations