Finding the Product of only 32 Numbers!

Algebra Level 4

Find the product of all numbers of the form ϵ 1 3 + ϵ 2 3 + ϵ 3 3 + ϵ 4 3 + ϵ 5 3 , \epsilon_1 \sqrt{3+\epsilon_2 \sqrt{3+\epsilon_3 \sqrt{3+\epsilon_4 \sqrt{3+\epsilon_5 \sqrt{3}}}}}, where ϵ k ∈ { − 1 , 1 } \epsilon_k \in \{-1, 1\} for each integer k k satisfying that 1 ≤ k ≤ 5. 1\leq k\leq 5.


The answer is 1179393.

Arturo Presa by Arturo Presa , 62, USA

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1 solution

Pi Han Goh
Jan 4, 2021

All 32 of these numbers satisfy the equation, x = ± 1 3 ± 2 3 ± 3 3 ± 4 3 ± 5 3 \large x = \pm_1 \sqrt{3 \pm_2 \sqrt{3 \pm_3 \sqrt{3 \pm_4 \sqrt{3 \pm_5 \sqrt3}}}}

By repeated squaring, we will get a ( 2 5 = 32 ) (2^5=32) degree polynomial, ( ( ( ( ( x 2 − 3 ) 2 − 3 ) 2 − 3 ) 2 − 3 ) 2 − 3 ) 2 = 0. (((((x^2 -3)^2 - 3)^2 - 3)^2 - 3)^2 - 3)^2 = 0. By Vieta's formula , the answer is ( ( ( 3 2 − 3 ) 2 − 3 ) 2 − 3 ) 2 − 3 = 1 179 393 . (((3^2-3)^2-3)^2-3)^2 - 3 = \boxed{\num{1179393}}.

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@Pi Han Goh , I deleted my solution because it was identical to yours.

Arturo Presa - 5 months, 1 week ago

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Not necessary. I love your questions by the way. Especially this one , which is giving me a huge headache.

Pi Han Goh - 5 months, 1 week ago

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Thank you, @Pi Han Goh

Arturo Presa - 5 months, 1 week ago

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