Question

sum of two number which can be expressed as the sum of two cubes the one who gives the answer first i will mark it as brainiest

Answer 1

Answer:

1729

Step-by-step explanation:

Ramanujan number that can be expressed as the sum of two cubes in two different ways. 1729 =12cube +1cube. 1729=10 cube+9 cube. Some other numbers are.

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Answer 2

Answer:

Hi Dear !

Step-by-step explanation:

Theorem: Let N be a positive integer. Then the equation N=x3+y3 has a solution in positive integers x,y if and only if the following conditions are satisfied:

There exists a divisor m|N with N1/3≤m≤(4N)1/3.

And m2−4m2−N/m3−−−−−−−−−−−−√ is an integer.

The sequence of integers F(n),

F(n)=a3+b3=(2n+6n2+6n3+n4)3+(n+3n2+3n3+2n4)3=c3+d3=(1+4n+6n2+5n3+2n4)3+(−1−4n−6n2−2n3+n4)3

for integer n>3 apparently is expressible as a sum of two positive integer cubes in exactly and only two ways.

F(4)F(5)F(60)=7443+7563=9453+153=15353+17053=20463+2043⋮=142777203+265788603=270218413+125061593

Using Broughan's theorem, I have tested F(n) from n=4−60 and, per n, it has only two solutions m, implying in that range it is a sum of two cubes in only two ways. Can somebody with a faster computer and better code test it for a higher range and see when (if ever) the proposed statement breaks down? Incidentally, we have the nice relations,

a+b=3n(n+1)3

c+d=3n3(n+1)

Note: F(60) is already much beyond the range of taxicab T3 which is the smallest number that is the sum of two positive integer cubes in three ways.

T3≈444.013=1673+4363=2283+4233=2553+4143

(Using the theorem, this yields 3 values for m.)