Not-so-simple addition

Number Theory Level 3

What is the smallest positive integer that cannot be represented as the sum of 18 or fewer biquadrates (fourth powers)?


The answer is 79.

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Denton Young by Denton Young , 47, USA

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1 solution

Denton Young
Dec 16, 2016

Any number less than 64 can be represented as (1, 2 or 3 copies of 2 4 2^4 ) plus at most 15 copies of 1 4 1^4 , so 18 or fewer biquadrates. 64 requires 4 copies of 2 4 2^4 . To reach 19 biquadrates, we need to add 15 copies of 1 4 1^4 . 64 + 15 = 79.

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Can you add details on what a "biquadrate" is? Do you mean "fourth powers"?

Calvin Lin Staff - 4 years, 6 months ago

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2nd power = "square", 3rd power = "cube", 4th power = "biquadrate". Standard mathematical notation.

Denton Young - 4 years, 6 months ago

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Thanks! I've added that in.

I don't think it's standard mathematical notation now, or at least I don't recall hearing about it till now. (Though, I've seen "biquadrate equation", defined as a x 4 + b x 2 + x = 0 ax^4 + bx^2 + x = 0 .)

Calvin Lin Staff - 4 years, 6 months ago

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