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Forum index » Science and Technology » Math » Recreational
Representation of Integers
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amcwill417
science forum addict


Joined: 04 May 2005
Posts: 65

PostPosted: Sun Jul 03, 2005 2:31 am    Post subject: Representation of Integers Reply with quote

Eah positive integer can be uniquely represented by three ordered smaller
integers per the following examples: 11 = (2,1,2), 18 = (2,3,1), 31 =
(3,2,0). What is the rule?

Alex
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Timothy Little
science forum Guru Wannabe


Joined: 30 May 2005
Posts: 295

PostPosted: Sun Jul 03, 2005 4:58 am    Post subject: Re: Representation of Integers Reply with quote

amcwill417 wrote:
Quote:
Eah positive integer can be uniquely represented by three ordered
smaller integers per the following examples: 11 = (2,1,2), 18 =
(2,3,1), 31 = (3,2,0). What is the rule?

Although there are infinitely many such rules, the first one that
sprang to mind worked:

N is represented by (a, b, c), where
a = floor(N ^ (1/3)),
b = floor((N - a^3) ^ (1/2)),
c = N - a^3 - b^2.

Or more colloquially, a is the cube root, b is the square root of
what's left, and c is what's left after that.

The reason why it sprang to mind is that I thought to myself that the
most compact such representation would asymptotically have
max(a,b,c) ~= N^(1/3).

If you take out the largest possible a^3, you'd have a remainder on
the order of 3a^2, so taking out a largest possible square would be
useful. Then you'd be left with a linear remainder.


- Tim
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amcwill417
science forum addict


Joined: 04 May 2005
Posts: 65

PostPosted: Mon Jul 04, 2005 1:25 am    Post subject: Re: Representation of Integers Reply with quote

"Timothy Little" <tim-usenet@little-possums.net> wrote in message
news:slrndces37.soe.tim-usenet@soprano.little-possums.net...
Quote:
amcwill417 wrote:
Eah positive integer can be uniquely represented by three ordered
smaller integers per the following examples: 11 = (2,1,2), 18 =
(2,3,1), 31 = (3,2,0). What is the rule?

Although there are infinitely many such rules, the first one that
sprang to mind worked:

N is represented by (a, b, c), where
a = floor(N ^ (1/3)),
b = floor((N - a^3) ^ (1/2)),
c = N - a^3 - b^2.

Or more colloquially, a is the cube root, b is the square root of
what's left, and c is what's left after that.

The reason why it sprang to mind is that I thought to myself that the
most compact such representation would asymptotically have
max(a,b,c) ~= N^(1/3).

If you take out the largest possible a^3, you'd have a remainder on
the order of 3a^2, so taking out a largest possible square would be
useful. Then you'd be left with a linear remainder.


- Tim

Yes. How do primes behave in this representation?

Alex
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amcwill417
science forum addict


Joined: 04 May 2005
Posts: 65

PostPosted: Mon Jul 04, 2005 4:39 pm    Post subject: Re: Representation of Integers Reply with quote

"amcwill417" <amcwill417@email.msn.com> wrote in message
news:b22ye.128$4H2.1394@eagle.america.net...
Quote:

"Timothy Little" <tim-usenet@little-possums.net> wrote in message
news:slrndces37.soe.tim-usenet@soprano.little-possums.net...
amcwill417 wrote:
Eah positive integer can be uniquely represented by three ordered
smaller integers per the following examples: 11 = (2,1,2), 18 =
(2,3,1), 31 = (3,2,0). What is the rule?

Although there are infinitely many such rules, the first one that
sprang to mind worked:

N is represented by (a, b, c), where
a = floor(N ^ (1/3)),
b = floor((N - a^3) ^ (1/2)),
c = N - a^3 - b^2.

Or more colloquially, a is the cube root, b is the square root of
what's left, and c is what's left after that.

The reason why it sprang to mind is that I thought to myself that the
most compact such representation would asymptotically have
max(a,b,c) ~= N^(1/3).

If you take out the largest possible a^3, you'd have a remainder on
the order of 3a^2, so taking out a largest possible square would be
useful. Then you'd be left with a linear remainder.


- Tim

Yes. How do primes behave in this representation?

Alex

What I have in mind here is this: 7 = (1,2,2), 11 = (2,1,2) and 13 =

(2,2,1) so that all permutations here yield primes. This is by no means
general but one can ask the question thus: Are there any other primes p =
(a,b,c) such that all permutations are also primes?

Alex
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