A180139 a(n)=A179387(n)+1.

4, 6, 33, 36, 38, 64, 66, 137, 569, 5216, 367807, 939788, 6369040, 7885439, 9536130, 140292678, 184151167, 890838664, 912903446, 3171881613

Offset: 1

Author

Artur Jasinski, Aug 12 2010

Keywords

• hard
• more
• nonn

Comments

Theorem (*Artur Jasinski*):

For any positive number x >= A180139(n) distance between cube of x and square of any y (such that x<>n^2 and y<>n^3) can't be less than A179386(n+1).

Proof: Because number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and completely computable, such x can't exist.

If x=n^2 and y=n^3 distance d=0.

For d values see A179386.

For y values see A179388.

Examples

			For numbers x from 4 to infinity distance can't be less than 4.
For numbers x from 6 to infinity distance can't be less than 7.
For numbers x from 33 to infinity distance can't be less than 26.
For numbers x from 36 to infinity distance can't be less than 28.
For numbers x from 38 to infinity distance can't be less than 49.
For numbers x from 66 to infinity distance can't be less than 60.
For numbers x from 137 to infinity distance can't be less than 63.
For numbers x from 569 to infinity distance can't be less than 174.
For numbers x from 5216 to infinity distance can't be less than 207.
For numbers x from 367807 to infinity distance can't be less than 307.
		

Crossrefs

Cf. A179107, A179108, A179109, A179387, A179388