%parodia.tex: The Sum of Two and Seven and The Product of Three by Three
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\centerline
{
\bf The Sum of Two and Seven and The Product of Three by Three
}
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{
\it Semaj Srelles
}
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({\eightrm To appear in Journal of Integers (JI) [accepted by J. Rodan, editor-in-chief]} )
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{\bf Abstract}

{\it In this brief note, we prove a result which was ``accidently''
found thanks to Lien Enaols's Online Encyclopedia of Integers. Namely,
we prove  by elementary techniques that the number that you get
when you add the integers two and seven equals the number that
you get when you multiply the integer three by itself.}



{\bf 1 Introduction}

Recently I received an electronic mail message (from Knarf Namrelle)
in which I was notified that a pair of duplicate entries existed in
Lien Enaols's On-Line Encyclopedia of Integers.
This involved entry 285100A, and the former entry 367300A.
One of these entries was described as the outcome of
{\it adding} the integers two and seven (i.e. $2+7$, in
technical jargon), while the other
was the {\it product} of the integer three by itself
(in other words $3 \times 3$).

I soon notified Lien Enaols of this fact and he promptly combined the
two entries into one entry, 285100A . Upon combining these two
entries, he also noted that it was not officially a theorem
(that adding two and seven 
{\it always} equals the product of three by itself),
but that it was ``certainly true''.

The goal of this short note is to prove this result so that its status
is indeed a theorem. We close the paper by noting two other
results relating addition and multiplication.


{\bf 2 The Results}

{\bf Theorem 2.1} The outcome of adding the integer two and the
integer seven equals the outcome of multiplying the integer
three by itself.

{\it Proof.}  In {\it base-one} (unary)
$$
2=11 \quad,
$$
$$
7=1111111 \quad .
$$
Hence
$$
2+7=11+1111111=111111111 \quad .
\eqno(1)
$$
Also
$$
3=111 \quad,
$$
Hence (recall that 3a=a+a+a)
$$
3 \times 3= 3+3+3=111+111+111=111111111 \quad.
\eqno(2)
$$
Since the right sides of Eqs. (1) and (2) are the same, so are
their left-sides (by the well-known {\it transitivity} of the $=$ relation).
\halmos

We note that at least two other entries, in OEI,
that came from addition, have elegant expressions as a product.
We first consider the entry 577300A.

{\bf Theorem 2.2} The outcome of adding the integer one and the
integer three equals the outcome of multiplying the integer
two by itself.

{\it Proof.} 
Tautologically we have
$$
1=1 \quad,
$$
and by definition (again, in {\it unary})
$$
3=111 \quad .
$$
Hence
$$
1+3=1+111=1111 \quad .
\eqno(3)
$$
Also
$$
2=11 \quad,
$$
Hence (recall that 2a=a+a)
$$
2 \times 2= 2+2=11+11=1111 \quad.
\eqno(4)
$$
Since the right sides of Eqs. (3) and (4) are the same, so are
their left-sides (by the well-known {\it transitivity}
of the $=$ relation).
\halmos

Finally we consider the entry 074820A. This leads to the conjencture

{\bf Conjecture}: The outcome of adding the integer seven and the
integer nine equals the outcome of multiplying the integer
four by itself. 

We leave the proof of this (assuming it is true) to the reader,
as the calculations involved herein would be straightforward
but tedious.

{\bf 3 Acknowledgements}

The author gratefully acknowledges Knarf Namrelle for bring this
problem to his attention and Rep Nakah Wodnul for valuable
e-conversations. The author also thanks the referees for
their helpful insights.




\end




