[seqfan] Re: Minimal k > n such that (4k+3n)(4n+3k) is square

[Susanne Wienand]

Dear seqfans,

other ratios of coprime (n,k) which fulfil (3k+4n)(4k+3n) = square number
can be appended to (1,393), (4,109), (13,132), (24,157) etc like dominoes:

(1,393), (393,1177), (1177,2353), (2353,3921), (3921,5881), (5881 8233)...

(4,109), (109,312), (312,613), (613,1012), (1012,1509), (1509,2104)...

(13,132), (132,349), (349,664), (664,1077), (1077,1588), (1588,2197)...

(24,157), (157,388), (388,717), (717,1144), (1144,1669), (1669,2292)...

The same numbers can be reached twice:

(321,7153), (7153,20257)

(769,3289), (3289,7377), (7377,13033), (13033,20257)...

Regards
Susanne







2013/12/26 Kevin Ryde <user42 at zip.com.au>

> charles.greathouse at case.edu (Charles Greathouse) writes:
> >
> > If there is no n < k < 109n/4 with (4k+3n)(4n+3k) square, then a(n) =
> 393n.
>
> I tried continuing past 393 where there's further multipliers
>
>     f=393, 76441, 14829361, 2876819793
>
> which is solutions to 12*f^2+25*f+12=square, excluding multiples of
> earlier solutions, and which therefore k=f*n gives (4k+3n)(4n+3k)=square
> for any n.
>
> It seems when a particular n allows f=109/4=27.25 that there's a single
> extra solution between the integer ones.  Eg. n=4
>
>     f = 109/4 = 27.2500
>         393
>         21949/4 = 5487.2500
>         76441
>         4258797/4 = 1064699.2500
>
> But when there's more than one solution they're in pairs.  Eg. n=13
>
>     f = 10.1538 = 132/13         \ new pair
>         70.4615 = 916/13         /
>         393
>         2170.4615 = 28216/13        \ new pair
>         13870.1538 = 180312/13      /
>         76441
>         421259.3846 = 5476372/13      \ new pair
>         2690939.3846 = 34982212/13    /
>         14829361
>
> It's possible to have a mixture of 27.25 and pairs.  Eg. n=24
>
>     f = 157/24 = 6.5417                <----+
>         654/24 = 27.2500    <- single       |- pair
>         2509/24 = 104.5417             <----+
>         393
>         35269/24 = 1469.5417
>         131694/24 = 5487.2500
>         491557/24 = 20481.5417
>         76441
>
> Dunno if this is always so or if it has any significance.
>
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