The Magnificent Seven

Submitted by Larry Engman, 12 December 1996. Original article by Valerio De Angelis, this article by Valerio De Angelis and Allen Stenger.

Find seven, positive, all different, whole numbers

x₁, x₂, x₃, x₄, x₅, x₆, x₇

such that

1/x₁ + 1/x₂ + 1/x₃ + 1/x₄ + 1/x₅ + 1/x₆ + 1/x₇ = 1

Extra credit: What if we had asked for six such numbers? ...or eight?

(Remark. Fractions with a 1 in the numerator are called unit fractions, or less commonly Egyptian fractions or Ahmes fractions. The ancient Egyptians did not have a fully-developed concept of fractions; instead they performed most fractional calculations using a representation of the fraction as a sum of unit fractions as we are doing here. This method is described in detail in an ancient document today called the Rhind papyrus (after its owner) or the Ahmes papyrus (after its scribe).)

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