For Carl Sagan (1934-1996), an astronomer, science popularizer,
novelist, and humanist:

1 + 5^k + (3+2y)^k + (3-2y)^k + (-3+3y)^k + (-3-3y)^k = (-2+x)^k + (-2-
x)^k + (5-y)^k + (5+y)^k

for k = 1,3,5,7 if x^2-10y^2 = 9.

This has, er, billions and billions (Sagan's favorite catchphrase) of
solutions.  Actually, there is an infinite number of rational
solutions to the conditional equation, as well as integral ones if it
is to be treated as a Pell equation.

Identities like these are in the context of the Prouhet-Tarry-Escott
Problem. It is easily shown that equations valid for k = 1,3,5,7 can
be tweaked (with an increase in the number of terms) such that it will
be valid for k = 1,2,3,...8.

I'll be creating the Wikipedia article for "Sagan's Identity" in a
while.

2. "X's" Identity: Selling an equation in eBay?

It seems naming conventions for scientific objects can be whimsical.
In biology, some species are named after movie actors: an ant for
Harrison Ford, a beetle for Schwarzenegger, etc. See:

http://www.mentalfloss.com/blogs/archives/16839

In astronomy, we have WIMPs (weakly interacting massive particles),
MACHOs (massive compact halo objects), etc. In genetics, we have the
"sonic hedgehog" gene (go figure), and others.

I was thinking of doing an experiment. I have another identity for k
=1,3,5,7 just like "Sagan's Identity". (In fact, both are special
cases of a more general one.)

Can I sell the right to name it in eBay?

The winning bidder can then name it after their favorite scientist (or
movie star, politician, or after himself/herself, as the case may be.
Certain exceptions/rules will apply: no Hitler, etc.).

After all, it is said, "Politics is for the moment, but an equation is
forever."

Just a thought I'm considering. (I wonder how much it would
go?)....   :-)

Yours,