Erdos Woods Numbers -- #A059756
Victor S. Miller
victor at idaccr.org
Sun Sep 4 17:24:45 CEST 2005
Here's a quick run down of the two ideas. Erdos had made the
following conjecture, which Woods investigated in his thesis:
Definition: If a is an integer and k is a positive integer. Define
P(a,k) = { primes dividing a,a+1,...,a+k }
Conjecture: There is an integer k, such that P(a,k) determines a
uniquely for all positive a.
Thus the definition that you saw of "Erdos Woods" pair.
In the course of looking at the conjecture Woods (foolishly)
conjectured that if m>1, the interval [a,a+m] always contains an
integer (not an end point) which is relatively prime to one of the end
points. It appears that he never tried to test the conjecture when he
wrote his thesis, but found the first counterexample of m=16 later.
There's not much around that I've been able to find (basically the
references in the OEIS). If you look at Nik Lygeros' web page
http://lygeros.org you'll find a bunch of miscellaneous results, and a
history of wrong conjectures, such as
1) All (non-trivial) Erdos Woods numbers are even. It turns out that
Erdos and Selfridge (in 1971) had the first counterexample to that
-- 903.
2) All even squares are Erdos Woods numbers. It turns out that 26^2
is a counterexample (there are lots of others).
And, I agree with you, 1 is trivially an Erdos Woods number, since the
open interval is empty.
Victor
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