My question is related to the Integer Relation Detection Problem which can be formulated as:
$a_1x_1 + a_2x_2 + \cdots + a_nx_n = 0$
Where $\forall_{a_i} a_i\in\mathbb{Z},a_i
There are a few algorithms to solve this problem, listed on the wikipedia page linked.
My question: are there algorithms for a solution to the same problem with the modification that:
$a_1x_1 + a_2x_2 + \cdots + a_nx_n = 1$
Or equivalently (I believe):
$a_1x_1 + a_2x_2 + \cdots + a_nx_n = b$
$b\in \mathbb R$ is a given.
I would love to see some reduction to an existing problem with a polynomial-time algorithm, such as Integer Relation Detection, or Simultaneous Integer Relation Detection. Another possibility is that this is a hard problem.