[seqfan] Re: Figurate Pythagorean triples

[Charles Greathouse]

The question is about Pythagorean triples made of squares and triangular
numbers, but I'd be interested to know if it's solvable with other
figurates. Of course everything is k-polygonal for large enough k so there
have to be some limits!

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Thu, Mar 20, 2014 at 5:59 AM, Jim Nastos <nastos at gmail.com> wrote:

> Re. Step 1: it is known that at most one of a primitive pyth triplet can be
> square.
>
> Step 2:
>
> There is a family of pythagorean triples of the form
> k^3, T(k^2-1), T(k^2) where k^3 is any cube and T(i) is the (i)th
> triangular number (sum from 1 to i).
>
> The example given is 4^3, T(15), T(16)
> Other examples are 5^3, T(24), T(25) = 125, 300, 325
> 2^3, T(3), T(4) = 8, 6, 10
> 3^3, T(8), T(9) = 27, 36, 45
>
> This holds for every k. Each of these triplets have k as a common factor,
> so they can't be candidates.
>
> The example given happens to use 64 which is both a square and a cube ...
> perhaps coincidentally, but I'm not sure what property is really being
> asked here (pyth triples made up of squares and triangulars? or any
> figurates?)
>
> J
>
>
>
>
> On Wed, Mar 19, 2014 at 1:02 PM, Allan Wechsler <acwacw at gmail.com> wrote:
>
> > Step 1: They can't all be square, because that would solve a^4 + b^4 =
> c^4.
> >
> >
> > On Wed, Mar 19, 2014 at 3:30 PM, Charles Greathouse <
> > charles.greathouse at case.edu> wrote:
> >
> > > Notice that all members of the Pythagorean triple 64, 120, 136 are
> > > triangular or square. A shy friend of mine asks: Can a primitive
> > > Pythagorean triple have the same property for its three sides?
> > >
> > > Charles Greathouse
> > > Analyst/Programmer
> > > Case Western Reserve University
> > >
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