> Prompted by a recent reference to the section (D11) on Egyptian > fractions in Guy's 'Unsolved Problems in Number Theory', I noticed > mention of this problem (paraphrased):
> Order the rationals a/b, 0 < a/b < 1, (a, b) = 1, by increasing a+b > and then by increasing a: 1/2, 1/3, 1/4, 2/3, 1/5, 1/6, 2/5, 3/4 ...
> What is the earliest such rational that cannot be expressed as the > sum of n unit fractions?
I'm sure you mean n _distinct_ unit fractions.
> I was surprised that so few results were known: 2/3, 4/5, 8/11, 16/17, > 77/79 cannot be expressed as the sum of 1, 2, 3, 4, 5 unit fractions > respectively. So I write a simplistic perl program, which has found > (after just over a mips-year of work) that 732/733 is the first in the > list that cannot be expressed as the sum of 6 unit fractions.
> Since this program exercises perl beyond the limits of its integer > accuracy, I cannot guarantee the result - can anyone confirm it for me?
I dusted off some old Egyptian fraction code I wrote a couple of years ago. It uses Common Lisp's exact integer arithmetic, so roundoff is not an issue. I verified that all fractions up to 732/733 have six-term representations and that 732/733 requires seven terms.
It took about 13 hours on a 296 Mhz Ultra. If a mips is a Mhz, that's almost half a mips-year. I don't how much of the remaining speed difference is due to the programming language/system and how much is due to the algorithm.
By the way, I found that 732/733 has 2771 different seven-term representations. The largest denominator appears in the representation (2305193137933140 33397845 4484 45 7 3 2). The smallest maximum denominator appears in (26388 20524 7330 45 7 3 2).