Unit fractions

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A unit fraction is a fraction having a numerator of 1. For example, $\scriptstyle {\frac {1}{7}}$ is a unit fraction.

Problems involving unit fractions

There are various problems in number theory involving unit fractions.

Smallest k such that 1/k can be written as a sum of exactly 2 unit fractions in n ways

The smallest integer $\scriptstyle k$ such that $\scriptstyle {\frac {1}{k}}$ can be written as a sum of two unit fractions in exactly $\scriptstyle n$ ways is given by

{1, 2, 4, 8, 6, 32, 64, 12, 256, 512, 24, 2048, 36, 30, 16384, 32768, 96, 72, 262144, 192, 1048576, 2097152, 60, 8388608, 216, 768, 67108864, 288, 1536, 536870912, ...} (A016017)

Erdős-Straus conjecture

The Erdős-Straus conjecture says that

${\frac {4}{n}}={\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}$

always has solutions in integers.

See also

• Index to fractions in OEIS
• Egyptian fractions
• Egyptian representations and greedy Egyptian representation
• Signed Egyptian representations and greedy signed Egyptian representation