[seqfan] Re: Unit fractions, distinct or not

[hv at crypt.org]

Super, thanks for the confirmation.

Clearly given q = sum_{s in S}{1/s} with d(q) = c(q), there will be some
n_0 such that q' = sum_{s in S \union {n}}{1/s} implies d(q') = c(q')
for all n >= n_0. I suspect that means it will be very hard to characterize
the set of q with d(q) = c(q).

For now I'll look for primitive S (such that 2 < q < 2 + 1/max(S))
with d(q) = c(q); maybe some structure will emerge.

Hugo

Lucas Brown <lucasbrown.cit at gmail.com> wrote:
: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.
:>
:> --
:> Seqfan Mailing list - http://list.seqfan.eu/
:>
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