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 enzocreti 2020-03-21 21:47

Diophantine equation

Are there infinitely many solutions to these Diophantine equation

10^n-a^3-b^3=c^2 with n, a, b, c positive integers?

 R. Gerbicz 2020-03-21 21:51

Yes.

 enzocreti 2020-03-21 21:54

[QUOTE=R. Gerbicz;540413]Yes.[/QUOTE]

How to proof it?

 VBCurtis 2020-03-21 23:00

Even wikipedia articles would give you sufficient tools to answer your curiosities.

 CRGreathouse 2020-03-22 04:46

[QUOTE=enzocreti;540412]Are there infinitely many solutions to these Diophantine equation

10^n-a^3-b^3=c^2 with n, a, b, c positive integers?[/QUOTE]

It seems rather difficult to find positive integers n such that there are no positive integers a, b, and c with 10^n = a^3 + b^3 + c^2. n = 5 is an example, but there are no others up to 18.

I don't know of any modular obstructions.

 R. Gerbicz 2020-03-22 08:39

[QUOTE=CRGreathouse;540436]It seems rather difficult to find positive integers n such that there are no positive integers a, b, and c with 10^n = a^3 + b^3 + c^2. n = 5 is an example, but there are no others up to 18.

I don't know of any modular obstructions.[/QUOTE]

Multiple the equation by 10^(6*k), since it is cube and square if there is a solution for n=N, then there is a solution for N+6*k.

 CRGreathouse 2020-03-23 03:03

[QUOTE=R. Gerbicz;540442]Multiple the equation by 10^(6*k), since it is cube and square if there is a solution for n=N, then there is a solution for N+6*k.[/QUOTE]

Ah! of course. So it suffices to show
10^1 = 1^3 + 2^3 + 1^2
10^2 = 3^3 + 4^3 + 3^2
10^3 = 6^3 + 7^3 + 21^2
10^4 = 4^3 + 15^3 + 81^2
10^6 = 7^3 + 26^3 + 991^2
10^11 = 234^3 + 418^3 + 316092^2
and then we know that all powers of 10, other than 10^5, are expressible as the sum of two positive cubes and a positive square.

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