The following is a well-known unsolved problem :(adsbygoogle = window.adsbygoogle || []).push({});

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.

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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?

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# Egyptian Fractions

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