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Are these numbers known?
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Steve Posner
Dec 4, 2011, 4:17:15 PM12/4/11
to
Are there pairs of distinct numbers (p, q) such that they are
right-left reversals of each other (e.g. 38 and 83) such that
their squares are also right-left reversals?
If so, do they have a name?
tia
sp
Andrew B
Dec 4, 2011, 5:30:47 PM12/4/11
to
On 04/12/2011 21:17, Steve Posner wrote:
>
>
> Are there pairs of distinct numbers (p, q) such that they are
> right-left reversals of each other (e.g. 38 and 83) such that
> their squares are also right-left reversals?
You mean like (12,21) and (13,31)?
>
>
> Are there pairs of distinct numbers (p, q) such that they are
> right-left reversals of each other (e.g. 38 and 83) such that
> their squares are also right-left reversals?
> If so, do they have a name?
Eric Sosman
Dec 4, 2011, 6:26:05 PM12/4/11
to
On 12/4/2011 4:17 PM, Steve Posner wrote:
>
> Are there pairs of distinct numbers (p, q) such that they are
> right-left reversals of each other (e.g. 38 and 83) such that
> their squares are also right-left reversals?
12, 21 -> 144, 441
>
> Are there pairs of distinct numbers (p, q) such that they are
> right-left reversals of each other (e.g. 38 and 83) such that
> their squares are also right-left reversals?
13, 31 -> 169, 961
102, 201 -> 10404, 40401
103, 301 -> 10609, 90601
112, 211 -> 12544, 44521
113, 311 -> 12769, 96721
122, 221 -> 14884, 48841
1002, 2001 -> 1004004, 4004001
1003, 3001 -> 1006009, 9006001
1012, 2101 -> 1024144, 4414201
1013, 3101 -> 1026169, 9616201
1022, 2201 -> 1044484, 4844401
...
> If so, do they have a name?
--
Eric Sosman
eso...@ieee-dot-org.invalid
gerson
Dec 5, 2011, 7:55:46 AM12/5/11
to
> On 12/4/2011 4:17 PM, Steve Posner
>> [that's firstly, and now famously]
>> wrote
>> Are there pairs of distinct numbers (p, q) such that they are
>> right-left reversals of each other (e.g. 38 and 83) such that
>> their squares are also right-left reversals?
>> Are there pairs of distinct numbers (p, q) such that they are
>> right-left reversals of each other (e.g. 38 and 83) such that
>> their squares are also right-left reversals?
>> If so, do they have a name?
>
and Eric Sosman said
>
>
> 12, 21 -> 144, 441
> 13, 31 -> 169, 961
>
> 102, 201 -> 10404, 40401
> 103, 301 -> 10609, 90601
> 112, 211 -> 12544, 44521
> 113, 311 -> 12769, 96721
> 122, 221 -> 14884, 48841
>
> 1002, 2001 -> 1004004, 4004001
> 1003, 3001 -> 1006009, 9006001
> 1012, 2101 -> 1024144, 4414201
> 1013, 3101 -> 1026169, 9616201
> 1022, 2201 -> 1044484, 4844401
> ...
> > They do now: the Posner numbers.
he did
! !
+ +
Now then, they might go on for ever, might seem so, but hmm the digits in these early ones look special -
If they go on forever, does that look change ?
Willem
Dec 5, 2011, 8:13:43 AM12/5/11
to
gerson wrote:
)> On 12/4/2011 4:17 PM, Steve Posner
)>> [that's firstly, and now famously]
)>> wrote
)
)>> Are there pairs of distinct numbers (p, q) such that they are
)>> right-left reversals of each other (e.g. 38 and 83) such that
)>> their squares are also right-left reversals?
)
)>> If so, do they have a name?
)>
)
) and Eric Sosman said
)
)>
)> 12, 21 -> 144, 441
)> 13, 31 -> 169, 961
)>
)> 102, 201 -> 10404, 40401
)> 103, 301 -> 10609, 90601
)> 112, 211 -> 12544, 44521
)> 113, 311 -> 12769, 96721
)> 122, 221 -> 14884, 48841
)>
)> 1002, 2001 -> 1004004, 4004001
)> 1003, 3001 -> 1006009, 9006001
)> 1012, 2101 -> 1024144, 4414201
)> 1013, 3101 -> 1026169, 9616201
)> 1022, 2201 -> 1044484, 4844401
)> ...
)> > They do now: the Posner numbers.
)
) he did
)
) ! !
)
) + +
)
) Now then, they might go on for ever, might seem so, but hmm the digits in
) these early ones look special -
)
) If they go on forever, does that look change ?
No idea, but it seems to hinge on there being no carry in the
multiplications, so one would guess it stops after a while.
Perhaps there are outliers that don't depend on no carry.
SaSW, Willem
--
Disclaimer: I am in no way responsible for any of the statements
made in the above text. For all I know I might be
drugged or something..
No I'm not paranoid. You all think I'm paranoid, don't you !
#EOT
)> On 12/4/2011 4:17 PM, Steve Posner
)>> [that's firstly, and now famously]
)>> wrote
)
)>> Are there pairs of distinct numbers (p, q) such that they are
)>> right-left reversals of each other (e.g. 38 and 83) such that
)>> their squares are also right-left reversals?
)
)>> If so, do they have a name?
)>
)
) and Eric Sosman said
)
)>
)> 12, 21 -> 144, 441
)> 13, 31 -> 169, 961
)>
)> 102, 201 -> 10404, 40401
)> 103, 301 -> 10609, 90601
)> 112, 211 -> 12544, 44521
)> 113, 311 -> 12769, 96721
)> 122, 221 -> 14884, 48841
)>
)> 1002, 2001 -> 1004004, 4004001
)> 1003, 3001 -> 1006009, 9006001
)> 1012, 2101 -> 1024144, 4414201
)> 1013, 3101 -> 1026169, 9616201
)> 1022, 2201 -> 1044484, 4844401
)> ...
)> > They do now: the Posner numbers.
)
) he did
)
) ! !
)
) + +
)
) Now then, they might go on for ever, might seem so, but hmm the digits in
) these early ones look special -
)
) If they go on forever, does that look change ?
No idea, but it seems to hinge on there being no carry in the
multiplications, so one would guess it stops after a while.
Perhaps there are outliers that don't depend on no carry.
SaSW, Willem
--
Disclaimer: I am in no way responsible for any of the statements
made in the above text. For all I know I might be
drugged or something..
No I'm not paranoid. You all think I'm paranoid, don't you !
#EOT
Ted Schuerzinger
Dec 5, 2011, 9:24:20 AM12/5/11
to
On Mon, 5 Dec 2011 13:13:43 +0000 (UTC), Willem wrote:
> No idea, but it seems to hinge on there being no carry in the
> multiplications, so one would guess it stops after a while.
> Perhaps there are outliers that don't depend on no carry.
It looks to me as though
> No idea, but it seems to hinge on there being no carry in the
> multiplications, so one would guess it stops after a while.
> Perhaps there are outliers that don't depend on no carry.
1 [insert arbitray number of 0's] 2 and
1 [insert arbitray number of 0's] 3
won't have any carries:
(10^n + 2) ^ 2 = 10^2n + 4 * 10^n + 4
(2*10^n + 1) ^ 2 = 4 * (10^2n) + 4 * 10^n + 1
Those should be the reverse of each other.
--
Ted S.
fedya at hughes dot net
Now blogging at http://justacineast.blogspot.com
Willem
Dec 5, 2011, 9:46:38 AM12/5/11
to
Ted Schuerzinger wrote:
) On Mon, 5 Dec 2011 13:13:43 +0000 (UTC), Willem wrote:
)
)> No idea, but it seems to hinge on there being no carry in the
)> multiplications, so one would guess it stops after a while.
)> Perhaps there are outliers that don't depend on no carry.
)
) It looks to me as though
) 1 [insert arbitray number of 0's] 2 and
) 1 [insert arbitray number of 0's] 3
)
) won't have any carries:
)
) (10^n + 2) ^ 2 = 10^2n + 4 * 10^n + 4
) (2*10^n + 1) ^ 2 = 4 * (10^2n) + 4 * 10^n + 1
)
) Those should be the reverse of each other.
Obvious. I stand corrected. (I blame the holiday season.)
) On Mon, 5 Dec 2011 13:13:43 +0000 (UTC), Willem wrote:
)
)> No idea, but it seems to hinge on there being no carry in the
)> multiplications, so one would guess it stops after a while.
)> Perhaps there are outliers that don't depend on no carry.
)
) It looks to me as though
) 1 [insert arbitray number of 0's] 2 and
) 1 [insert arbitray number of 0's] 3
)
) won't have any carries:
)
) (10^n + 2) ^ 2 = 10^2n + 4 * 10^n + 4
) (2*10^n + 1) ^ 2 = 4 * (10^2n) + 4 * 10^n + 1
)
) Those should be the reverse of each other.
Obvious. I stand corrected. (I blame the holiday season.)
Duncan Booth
Dec 5, 2011, 10:10:14 AM12/5/11
to
Ted Schuerzinger <fe...@hughes.spam> wrote:
> On Mon, 5 Dec 2011 13:13:43 +0000 (UTC), Willem wrote:
>
>> No idea, but it seems to hinge on there being no carry in the
>> multiplications, so one would guess it stops after a while.
>> Perhaps there are outliers that don't depend on no carry.
>
> It looks to me as though
> 1 [insert arbitray number of 0's] 2 and
> 1 [insert arbitray number of 0's] 3
>
> won't have any carries:
>
> (10^n + 2) ^ 2 = 10^2n + 4 * 10^n + 4
> (2*10^n + 1) ^ 2 = 4 * (10^2n) + 4 * 10^n + 1
>
> Those should be the reverse of each other.
>
There are other patterns which look like they are going to continue but
> On Mon, 5 Dec 2011 13:13:43 +0000 (UTC), Willem wrote:
>
>> No idea, but it seems to hinge on there being no carry in the
>> multiplications, so one would guess it stops after a while.
>> Perhaps there are outliers that don't depend on no carry.
>
> It looks to me as though
> 1 [insert arbitray number of 0's] 2 and
> 1 [insert arbitray number of 0's] 3
>
> won't have any carries:
>
> (10^n + 2) ^ 2 = 10^2n + 4 * 10^n + 4
> (2*10^n + 1) ^ 2 = 4 * (10^2n) + 4 * 10^n + 1
>
> Those should be the reverse of each other.
>
then break down when they hit a carry:
For example, 12, 121, 1211, 12111, 121111, 1211111 all work but 12111111
doesn't. I think that means although there is no upper limit to posner
numbers the highest one with no zeroes must be 2111111.
--
Duncan Booth http://kupuguy.blogspot.com
Peter Webb
Dec 5, 2011, 6:45:19 PM12/5/11
to
"Duncan Booth" <duncan...@invalid.invalid> wrote in message
news:Xns9FB299E2A5...@127.0.0.1...
peter wanker
Dec 7, 2011, 9:32:09 AM12/7/11
to
hit me when my car reached 114041 miles!
(12, 21) and (13, 31) came easily, but then a friend pointed out
the 'zero stuffed' numbers derived from these. As others have
done, it is trivial to prove that there are an infinite number of
them. Then, for fun, I tried the 'one stuffed' numbers, not
expecting them to work. Much to my disgust (!) (112, 211)
work. I gave up then as they seemed too common place -- not
so rare at all. But, I like the extension to see which families
terminate and which go on forever.
sp
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