# Existence of solutions to (n,n+1,n+1)

##
THEOREM (Jaroslaw Wroblewski, July 7, 2002):

For any n there exists a solution to **(n,n+1,n+1)**. However it may
have some (but not all!!!) of left terms equal to some of right terms,
so in fact it may reduce to a solution to **(n,k,k)** for a **k<=n+1**.

##
PROOF:

Let N be a large positive integer to be specified later.
Consider all sytems of positive integers
**(a**_{1},a_{2},...,a_{n+1})
such that
**N>=a**_{1}>=a_{2}>=...>=a_{n+1}.
There are at least
**s = N**^{n+1} / (n+1)!
such systems, as there are **N**^{n+1}
(n+1)-tuplets of positive
integers not exceeding N, and any nonincreasing sequence corresponds
to at most **(n+1)!** such tuplets.
For each of the systems, consider the sum
**a**_{1}^{n} + a_{2}^{n} + ... + a_{n+1}^{n}
being a number not exceeding
**S = (n+1) * N**^{n} .
For large N (namely **N>(n+1)*(n+1)!**) we have
**s > S**
and there are more considered systems than possible sums associated
with them. By pigeon-hole principle there exist two different systems
having the same sum associated with them, giving
** a**_{1}^{n} + a_{2}^{n} +
... + a_{n+1}^{n} =
b_{1}^{n} + b_{2}^{n} +
... + b_{n+1}^{n}
with
**N>=a**_{1}>=a_{2}>=...>=a_{n+1} and
**N>=b**_{1}>=b_{2}>=...>=b_{n+1}.
Although systems
**(a**_{1},a_{2},...,a_{n+1}) and
**(b**_{1},b_{2},...,b_{n+1}) are
different, they may have some common elements.

This ends the proof of the theorem.
##
REMARKS:

Proof is not constructive at all. It proves there exists a solution to **(n,n+1,n+1)**, but gives no indication how to find a particular example.

As a corollary we get the following (the first place where the estimates in power table are proven to be improvable):

There exists a solution to **(24,25,25)**.
It may have some (but not all) common terms on left and right sides. Such solution involves numbers not exceeding
**25*25!+1=387780251083274649600000001** (27 digits)

Jaroslaw Wroblewski, July 7, 2002
##
EXTENSION:

We can extend Jaroslaw Wroblewski's proof to show that there exist
linked solutions to

**a**_{1,1}^{n} + a_{1,2}^{n} + ... + a_{1,n+1}^{n}
= a_{2,1}^{n} + a_{2,2}^{n} + ... + a_{2,n+1}^{n
}= ... = a_{m,1}^{n} + a_{m,2}^{n}
+ ... + a_{m,n+1}^{n}
for arbitrarily large **m**.

The number of systems **s = N**^{n+1} / (n+1)! and the number
of sums** S = (n+1) * N**^{n} are as before.

If the number of systems is more than **m-1 **times the number of sums,
then by the pigeon-hole principle there is at least one sum
that must have **m** systems associated with it. This occurs when **s>(m-1)*S
**or **N>(m-1)*(n+1)*(n+1)!**

Stuart Gascoigne, August 5, 2002

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