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The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above.
Simpler lower and upper bounds for Frobenius numbers for n = 3 have also been determined. The asymptotic lower bound due to Davison The asymptotic lower bound due to Davison f ( a 1 , a 2 , a 3 ) ≡ g ( a 1 , a 2 , a 3 ) + a 1 + a 2 + a 3 ≥ 3 a 1 a 2 a 3 {\displaystyle f(a_{1},a_{2},a_{3})\equiv g(a_{1},a_{2},a_{3})+a_{1}+a_{2}+a_{3}\geq ...
The exact MILP results for 3,4,5,6,7 correspond to the lower bound. For k>7, no exact results are known, but the difference between the lower and upper bound is less than 0.3%. When the parameter is the number of subsets (m), the approximation ratio is exactly .
The best known upper bound on the size of a square-difference-free set of numbers up to is only slightly sublinear, but the largest known sets of this form are significantly smaller, of size . Closing the gap between these upper and lower bounds remains an open problem.
The construction follows a recursion by starting with any number , that is not an upper bound (e.g. =, where and an arbitrary upper bound of ). Given I n = [ a n , b n ] {\displaystyle I_{n}=[a_{n},b_{n}]} for some n ∈ N {\displaystyle n\in \mathbb {N} } one can compute the midpoint m n := a n + b n 2 {\displaystyle m_{n}:={\frac {a_{n}+b_{n ...
The lower bound was given by an easy argument. The upper bound is given by a n × n {\displaystyle {\sqrt {n}}\times {\sqrt {n}}} square grid. For such a grid, there are O ( n / log n ) {\displaystyle O(n/{\sqrt {\log n}})} numbers below n which are sums of two squares, expressed in big O notation ; see Landau–Ramanujan constant .
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