Abstract
Let 1 ≤ p < ∞ be any fixed real. We show that assuming P ≠ N P , it is hard to approximate the Minimum Solutions of Linear Diophantine Equations in ℓ p norm within any constant factor and it is also hard to approximate the Minimum Solutions of Linear Diophantine Equations in ℓ p norm within the factor n c / log log n for some constant c > 0 where n is the number of variables in the equations.
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