[seqfan] Re: Figurate Pythagorean triples

[Jim Nastos]

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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