Sierpinski Egyptian Fraction Conjecture

From: George Barwood (george.barwood.removethis_at_blueyonder.co.uk)
Date: 06/07/04 

Date: Mon, 7 Jun 2004 18:20:42 +0100

According to

http://planetmath.org/encyclopedia/SierpinskiErdosEgyptianFractionConjecture.html

and

http://mathworld.wolfram.com/EgyptianFraction.html

Sierpinski conjectured that for any integer n > 3 there exist positive
integers a,b,c such that

5/n = 1/a + 1/b + 1/c

Algebra shows that the equation is easily solved except for when n is of
form 10t+1.

I have written a simple program to calculate solutions for that case, which
indicates that there is no solution for n = 541, and thus the conjecture
appears to be false!

Naturally I suspect my program to be incorrect ( it hardly seems credible
that such a small counter-example can exist ).

Can anyone exhibit a solution for n=541?

Just in case it is of interest, my program is shown below.

{
  t : 1
  while t < 300
  {
    n : t * 10 + 1
    x : t * 2 + 1
    y : 4
    more : true
    while more
    {
      p : n * x
      b : 1
      c : y - b
      while Not( b > c )
      {
        if p % b = 0 && p % c = 0
        {
          Put( `<p>n=`, n )
          Put( ` x=`, x )
          Put( ` y=`, y )
          Put( ` b=`, b )
          Put( ` c=`, c )
          more = false
          b = c + 1
        }
        else
        {
          b = b + 1
          c = c - 1
        }
      }
      x = x + 1
      y = y + 5
      if x * 5 > 3 * n
      {
        Put( `<p>No solution when n=`, n )
        more = false
      }
    }
    t = t + 1
  }
}

Relevant Pages

  • Re: Figuring out all the subrings of this field!
    ... > I was trying to make some nice examples for myself in real algebra so I ... > want to first find out all the possible subrings of the field K, ... You are incorrect. ... R has a *LOT* of subfields, ...
    (sci.math)
  • Re: JSH: Assocation does not prove
    ... > Come on James, let's see some irrelevant, incoherent, and incorrect ... Yeah, bring it on! ...
    (sci.math)