# Smallest Positive Integer with 5 Fibonacci Partitions

## Theorem

The smallest positive integer which can be partitioned into distinct Fibonacci numbers in $5$ different ways is $24$.

## Proof

 $\displaystyle 1$ $=$ $\displaystyle 1$ $1$ way $\displaystyle 2$ $=$ $\displaystyle 2$ $1$ way $\displaystyle 3$ $=$ $\displaystyle 3$ $\displaystyle$ $=$ $\displaystyle 2 + 1$ $2$ ways $\displaystyle 4$ $=$ $\displaystyle 3 + 1$ $1$ way $\displaystyle 5$ $=$ $\displaystyle 5$ $\displaystyle$ $=$ $\displaystyle 3 + 2$ $2$ ways $\displaystyle 6$ $=$ $\displaystyle 5 + 1$ $\displaystyle$ $=$ $\displaystyle 3 + 2 + 1$ $2$ ways $\displaystyle 7$ $=$ $\displaystyle 5 + 2$ $1$ way $\displaystyle 8$ $=$ $\displaystyle 8$ $\displaystyle$ $=$ $\displaystyle 5 + 3$ $\displaystyle$ $=$ $\displaystyle 5 + 2 + 1$ $3$ ways $\displaystyle 9$ $=$ $\displaystyle 8 + 1$ $\displaystyle$ $=$ $\displaystyle 5 + 3 + 1$ $2$ ways $\displaystyle 10$ $=$ $\displaystyle 8 + 2$ $\displaystyle$ $=$ $\displaystyle 5 + 3 + 2$ $2$ ways $\displaystyle 11$ $=$ $\displaystyle 8 + 3$ $\displaystyle$ $=$ $\displaystyle 8 + 2 + 1$ $\displaystyle$ $=$ $\displaystyle 5 + 3 + 2 + 1$ $3$ ways $\displaystyle 12$ $=$ $\displaystyle 8 + 3 + 1$ $1$ way $\displaystyle 13$ $=$ $\displaystyle 13$ $\displaystyle$ $=$ $\displaystyle 8 + 5$ $\displaystyle$ $=$ $\displaystyle 8 + 3 + 2$ $3$ ways $\displaystyle 14$ $=$ $\displaystyle 13 + 1$ $\displaystyle$ $=$ $\displaystyle 8 + 5 + 1$ $\displaystyle$ $=$ $\displaystyle 8 + 3 + 2 + 1$ $3$ ways $\displaystyle 15$ $=$ $\displaystyle 13 + 2$ $\displaystyle$ $=$ $\displaystyle 8 + 5 + 2$ $2$ ways $\displaystyle 16$ $=$ $\displaystyle 13 + 3$ $\displaystyle$ $=$ $\displaystyle 13 + 2 + 1$ $\displaystyle$ $=$ $\displaystyle 8 + 5 + 3$ $\displaystyle$ $=$ $\displaystyle 8 + 5 + 2 + 1$ $4$ ways $\displaystyle 17$ $=$ $\displaystyle 13 + 3 + 1$ $\displaystyle$ $=$ $\displaystyle 8 + 5 + 3 + 1$ $2$ ways $\displaystyle 18$ $=$ $\displaystyle 13 + 5$ $\displaystyle$ $=$ $\displaystyle 13 + 3 + 2$ $\displaystyle$ $=$ $\displaystyle 8 + 5 + 3 + 2$ $3$ ways $\displaystyle 19$ $=$ $\displaystyle 13 + 5 + 1$ $\displaystyle$ $=$ $\displaystyle 13 + 3 + 2 + 1$ $\displaystyle$ $=$ $\displaystyle 8 + 5 + 3 + 2 + 1$ $3$ ways $\displaystyle 20$ $=$ $\displaystyle 13 + 5 + 2$ $1$ way $\displaystyle 21$ $=$ $\displaystyle 21$ $\displaystyle$ $=$ $\displaystyle 13 + 8$ $\displaystyle$ $=$ $\displaystyle 13 + 5 + 3$ $\displaystyle$ $=$ $\displaystyle 13 + 5 + 2 + 1$ $4$ ways $\displaystyle 22$ $=$ $\displaystyle 21 + 1$ $\displaystyle$ $=$ $\displaystyle 13 + 8 + 1$ $\displaystyle$ $=$ $\displaystyle 13 + 5 + 3 + 1$ $3$ ways $\displaystyle 23$ $=$ $\displaystyle 21 + 2$ $\displaystyle$ $=$ $\displaystyle 13 + 8 + 2$ $\displaystyle$ $=$ $\displaystyle 13 + 5 + 3 + 2$ $3$ ways $\displaystyle 24$ $=$ $\displaystyle 21 + 3$ $\displaystyle$ $=$ $\displaystyle 21 + 2 + 1$ $\displaystyle$ $=$ $\displaystyle 13 + 8 + 3$ $\displaystyle$ $=$ $\displaystyle 13 + 8 + 2 + 1$ $\displaystyle$ $=$ $\displaystyle 13 + 5 + 3 + 2 + 1$ $5$ ways

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