covering all N with squares (2nd degree polynomials with integer coeffs.).

[Simon Plouffe]

Hello,

  the title is a little short and not fully descriptible,

Here is the problem.

How to cover all N (natural numbers) with simple
poynomials of the second degree even with some overlaps.

We all know that it is possible to do it with a
beatty sequence when 1/a + 1/b = 1, with a,b irrationals.
[a*n] and [b*n] covers N with no overlab and no holes.

if a = 1/2 + sqrt(5)/2 and b = 0.38196601125 then it works.

Of course it is possible to do it with 2 or more ordinary
arithmetical progressions.

But how to do it with 2nd degree polynomials?

Is it possible?

For example (a tentative sieve that does not work),

n^2 =   1, 4, 9, 16, 25, ...
n^2+1 = 2, 5, 10, 17, 26, ...
n^2+2 = 3, 6, 11, 18, 27, ...
etc

does cover many integers but I do not think it does work.
It does not matter if there are some overlaps.

of course, in the above example, n^2+k at n=1 will eventually
reach any number but let's say : can it be done non-trivially?

I am just wondering if the problem has a solution.

simon plouffe