Finite Pi

Number Theory Level 3

There exists a finite sequence of integers { a k } k = 1 n \left \{ a_k \right \}_{k=1}^{n} such that

π = a 1 + 1 a 2 + 1 a 3 + 1 a 4 + 1 a 5 + 1 ⋱ a n \pi = a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{ a_4+\cfrac{1}{a_5+\cfrac{1}{\ddots a_n}}}}}

True False

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1 solution

Kb E
Nov 3, 2017

If there was such a sequence of integers, it could be simplified into a fraction π = a b \pi = \frac{a}{b} for a , b ∈ Z + a,b\in \mathbb{Z}^{+} . As π \pi is an irrational (and transcendenal) number, this is not possible.

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