>
> Friends:
>
> On page 3 of W S Anglin's
>
> Mathematics: A Concise History and Philosophy,
>
> there is the statement that
>
> In 1880 J J Sylvester proved that any proper fraction can be expressed
> as a sum of distinct unit fractions.
>
> Then Angelin gives a very brief and easy to follow proof by mathematical
> induction, which, I suppose is approximately the proof that Sylvester gave.
>
> Angelin doesn't claim that Sylvester's is the first proof, and I would be
> surprised to find that it hadn't been proved until 1880. Where by the way
> do we find Sylvester's proof?
>
> Best wishes,
>
> Sam Kutler
>
Fibonacci gives eight procedures to express fractions into sums of unit
fractions (singularis pars). The first six work only in special cases.
Procedures seven and eight work in all cases. His proofs are lacunary, but
can be turned into proofs that are acceptable to us. The seventh procedure
is of the greedy type: Given natural numbers a and b such that a < b. Determine
q and r such that $b = q.a + r and r < a. (The case that a divides b is trivial.
Fibonacci as a good mathematician does not forget to mention this situation.
It is useful to know in order to see when an algorithm terminates.) Then
1/(q + 1) < a/b < 1/q
and
a/b = 1/(q + 1) + (a - r)/((q + 1)b).
1/(q + 1) is the largest unit fraction below a/b. (Boncompagni edition 81--83.)
The algorithm takes at most a steps. This bound is best possible as the
example a and b = k.a! + 1 shows (k any integer). This, of course, is not
Fibonacci's remark. In general, the algorithm does not give the shortest
decomposition of a/b as a sum of unit fractions.
Best regards, Heinz Lueneburg