Solving the Unsolved: Egyptian Fractions

[LeoYard]

The following is a well-known unsolved problem :

If n is an integer larger than 1, must there be integers x, y, and z, such that 4/n=1/x+1/y+1/z?
A number of the form 1/x where x is an integer is called an Egyptian fraction.
Thus, we want to know if 4/n is always the sum of three Egyptian fractions, for n>1.
..................
So, if 4/n=1/x + 1/y + 1/z, then there exists a third degree polynomial x^3 + ax^2 + bx + c where 4/n=b/c for integers b and c. This is because 1/x + 1/y + 1/z=(z(x+y) +xy)/xyz. And the polynomial (a+x)(a+y)(a+z)=a^3 + (x+y+z)a^2 + (z(x+y)+xy)a +xyz, which proves the statement when b=z(x+y) +xy and c=xyz.

Your thoughts?

 

[robert Ihnot]

What about 1/2 +1/5 +1/7 = 59/70, how does that work out?

You are confused, as I read you, about The Erdős–Straus conjecture:
The form 4/N can always be expanded into three Egyptian fractions.

However, there has got to be restrictions on N, which Wikipedia puts at $$N \geq 2$$ and others just ignore. However, 2 will not work: 2 = 1/x + 1/y +1/z, if we are speaking of positive all different intergers for x,y,z. Thus the first acceptable case is 4/3 = 1+1/4+1/12, and then 4/4 = 1/+1/3+1/6; and 4/5 = 1/2+1/5+1/10; 4/6 =2/3=1/3+1/4+1/12; 4/7 =1/2+1/15 + 1/210. Maybe Erdos, etc, were assuming 4/N was less than 1 anyway.

 

[tacman]

It is interesting to see how the concept of Egyptian fractions can be applied to solve this unsolved problem. The use of a third degree polynomial to prove the statement is a clever approach and it shows the connection between Egyptian fractions and algebraic equations. However, it is important to note that this solution only works for integers b and c, and it does not necessarily prove that 4/n can always be expressed as the sum of three Egyptian fractions for all values of n. Further investigation and proof is needed to fully solve this problem. Nonetheless, this approach is a great step towards finding a solution and it highlights the potential of using mathematical concepts in unexpected ways to solve complex problems.