[seqfan] Re: Unit fractions, distinct or not

[Lucas Brown]

I once wrote a Python package that includes a function to generate all of
the shortest Egyptian fraction representations (that is, sums of unit
fractions with distinct denominators) of a rational number.  Running it
under PyPy3 on my computer, it took 23 seconds to determine that 1759 /
84882 has 1771 four-term EFRs and no shorter EFRs, and almost instantly
determined that 2 + 1759 / 84882 has only the one six-term EFR and no
shorter EFRs.

The software in question can be examined at
https://github.com/lucasaugustus/labmath/blob/main/labmath.py, or installed
via pip3 install labmath.  To use it, run the following code:

>>>> from labmath import egypt_short
>>>> for (n, x) in egypt_short(1759, 84882), start=1): print(n, x)
1 (49, 3178, 16859688, 378998755750104)
2 (49, 3178, 16859689, 162428047812723)
3 (49, 3178, 16859698, 26441789340414)
4 (49, 3178, 16859699, 24191425725033)
(lines suppressed)
1768 (105, 132, 276, 7669695)
1769 (105, 138, 253, 7669695)
1770 (112, 140, 215, 3395280)
1771 (129, 140, 172, 70735)
>>>> for (n, x) in enumerate(egypt_short(1759 + 2*84882, 84882), start=1):
print(n, x)
1 (1, 2, 3, 7, 43, 47)
>>>>

On Wed, Nov 1, 2023 at 8:27 AM <hv at crypt.org> wrote:

> A few years ago I found a nice proof [1] that the least number of unit
> fractions needed to sum to a rational q < 2 was the same whether or
> not you required the unit fractions to be distinct. At the time
> I considered also q > 2, but I could not prove that the distinct
> requirement always needed more unit fractions in these cases, nor
> could I find a counterexample.
>
> This was prompted by a couple of sequences in which it wasn't clear
> whether unit fractions were required to be distinct, and trying to
> work out whether or when it mattered.
>
> Looking at it again today, I quickly found this apparent counterexample:
>   1 + 1/2 + 1/3 + 1/7 + 1/43 + 1/47
>     = 171523/84882
>     = 2 + 1759/84882
> .. where it appears that 1759/84882 cannot be expressed as a sum of fewer
> than 4 unit fractions (eg 1/49 + 1/3178 + 1/16859688 + 1/378998755750104).
> So either last time or this time I made a mistake, and I'm not sure which.
>
> Given sets S and multisets M of positive integers,
> let d(q) := min(| S |): q = sum_{s \in S}{1/s}
> and c(q) := min(| M |): q = sum_{m \in M}{1/m}.
>
> I have two questions:
> - am I correct that q = 2 + 1759/84882 has d(q) = c(q) = 6?
> - can we say anything about the set of rationals q > 2 for which d(q) =
> c(q)?
>
> Hugo
>
> [1] Repeatedly apply a splitting function 2/(2k+1) = 1/(k+1) +
> 1/(k+1)(2k+1)
> to remove duplicates, and show that each step reduces the sum of a weight
> function w(n) defined as the greatest power of 2 dividing n-1, and must
> therefore terminate. This can show d(q) = c(q) for all rational 0 < q < 2.
>
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