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.
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