One to rule three cubes

Number Theory Level 5

Let the equation 1 = ( 1 + 9 m 3 ) 3 + ( 9 m 4 ) 3 + ( − 9 m 4 − m x ) 3 1 = (1+9m^3)^3 + (9m^4)^3 + (-9m^4-mx)^3 have solutions for all m ∈ N m \in \mathbb{N} and a certain x ∈ N x \in \mathbb{N} .

What is the smallest positive integer x x for this to be true?


The answer is 3.

Alisa Meier by Alisa Meier , 24, Germany

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Let

(1) f ( m , x ) = − 1 + ( 1 + 9 m 3 ) 3 + ( 9 m 4 ) 3 + ( − 9 m 4 − m x ) 3 . f(m,x) = -1 + (1+9m^3)^3 + (9m^4)^3 + (-9m^4-mx)^3.

Expanding and simplifying, one obtains

(2) f ( m , x ) = m 3 ( 3 − x ) [ 9 + 3 x + x 2 + 27 m 3 ( 3 + x ) + 243 m 6 ] . f(m,x) = m^3(3-x)[9+3x+x^2 + 27m^3(3+x) + 243m^6].

It follows from (2) that f ( m , 3 ) = 0 f(m,3) = 0 for all m ∈ N m\in\N . (Remark: such identity shows that there is an infinite number of ways of representing 1 as a sum of three cubes.)

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Chakravarthy B
Jan 27, 2019

Solution:m = 3

Refer tattersall-Elementary number theory in nine chapters book page no. 251, 11 th line

In 1936,K.Mahler discovered this identity

0 Helpful 0 Interesting 0 Brilliant 0 Confused

you have copied this from other source

chakravarthy b - 2 years, 3 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...