## Sunday, October 31, 2010

### My Favorite Unsolved problem: (3) Egyptian Fraction

A fraction written as a sum of distinct unit fractions is called an Egyptian Fraction.

E.g.

3/4 = 1/2 + 1/4

6/7 = 1/2 + 1/3 + 1/42

5/8 = 1/2 + 1/8

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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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.

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Here's a much easier problem:

(1) 1/a + 1/b + 1/c + 1/d + 1/e + 1/f = 1

(2) 1/a + 1/b + 1/c + 1/d + 1/e + 1/f + 1/g = 1

(3) 1/a + 1/b + 1/c + 1/d + 1/e + 1/f + 1/g + 1/h = 1

Find a, b, c, d, e, f, g, h

They are all positive, different, whole numbers.