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From: jm bergot on 14 Jun 2010 09:16
One notices that 36 can find two other squares:
(a)36=4*9 and 36+4+9=49
(b)36=2*18 and 36-(2+18)=16.

Is 36 unique? What other numbers can find two
squares in this manner of either adding or subtracting
pairs of multiplicans equaling the number?
From: Gerry Myerson on 14 Jun 2010 19:54
In article
<1672446060.332268.1276535831233.JavaMail.root(a)gallium.mathforum.org>,
jm bergot <thekingfishb(a)yahoo.ca> wrote:

> One notices that 36 can find two other squares:
> (a)36=4*9 and 36+4+9=49
> (b)36=2*18 and 36-(2+18)=16.
>
> Is 36 unique? What other numbers can find two
> squares in this manner of either adding or subtracting
> pairs of multiplicans equaling the number?

132 + 4 + 33 = 13^2, 132 - 2 - 66 = 8^2.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Tim Little on 15 Jun 2010 00:54
On 2010-06-14, Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> wrote:
> 132 + 4 + 33 = 13^2, 132 - 2 - 66 = 8^2.

There appear to be 50 values up to 10^6 that have this behaviour. The
only square among them appears to be 36. Searching among squares
specifically, I didn't find any others up to 200 000^2. I had a bit
of a stab at proving or refuting the conjecture that 36 is the only
square with this property, but with no conclusion yet.


- Tim
From: Gerry Myerson on 15 Jun 2010 01:04
In article <slrni1e1s8.jrj.tim(a)soprano.little-possums.net>,
Tim Little <tim(a)little-possums.net> wrote:

> On 2010-06-14, Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> wrote:
> > 132 + 4 + 33 = 13^2, 132 - 2 - 66 = 8^2.
>
> There appear to be 50 values up to 10^6 that have this behaviour.

Cool. Any patterns? Suggestions of infinite families?

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: jm bergot on 15 Jun 2010 08:20
MEGAthsnx to those who explored this item.
One can conclude that 36 is indeed gorgeous.
Somewhere I have a collection of other gorgeous
numbers and maybe will find time to send them
along
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