In Beauville's "Counting rational curves on K3 surfaces" is implictly
assumed the existence of algebraic K3 surfaces with Pic of rank one and generated 
by a curve of genus g. 

How do we show the existence of such K3 surfaces ? 
**Edit**: See Ferreti's first comment below for an answer. 

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Using the argument pointed out  by Ferreti in his first and second comments, we can
reduce the existence of  the sought K3 surfaces to the following: 

>   There exists infinitely many integers
> k for which (2k)(2k +3) is 
> squarefree.

Start with a smooth quartic S in P(3) and let H be a hyperplane section. 
For a fixed r and k>>0,  the restriction of kH to S is r-very ample. 

Suppose S contains a line L. Then the linear system |E|=|H-L| 
defines a fibration by elliptic curves on S. 
Thus kH + E is also r-very ample.

Let SS be a family of K3 surfaces 
that  deforms S in such a way that the class of O(k) is preserved, 
and for a generic member of the family every  element in H1,1\cap
H^2(Z) not proportional to O(k) becomes non-rational. Thus the 
generic element has Pic = Z. Since r-very ampleness is an open condition
( the points in the relative Hilb^r(SS)  where it does not hold is closed)
we obtain a K3 surface with Pic = Z  and  a r-very ample line-bundle of 
self-intersection 4k^2 + 6k = 2k(2k +3). If this  number is squarefree then
the line bundle is primitive. 


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After googling a bit I found general results about squarefree values of polynomials  which seems to ensure the existence of infinitely many integers k for which 2k(2k +3) is squarefree.  

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**Edit:** It is not clear to me  if it is really necessary to impose the number theoretical condition to obtain primitiviness.