Clear[CarryQ]
CarryQ[a_Integer, b_Integer] :=
Length@IntegerDigits@(a b) > Max@(Length@*IntegerDigits /@ {a , b})
SeedRandom[2];
list = RandomInteger[{20}, {5, 2}]~Join~RandomInteger[{1, 10}, {5, 2}]
{#1, #2, #1 #2, CarryQ[#1, #2]} & @@@ list //
Grid[#, Alignment -> Left] &
For the binary case, you can make minor changes given that IntegerDigits has an argument for base and len. If you want to use 2s-complement arithmetic, then you can preferably add more context about the task that requires this calculation.
EDIT (for the binary case)
Let's introduce register length reg that I can arbitrarily set at 8 for the default case; although the function takes it as a parameter. If the sum can fit in the assigned register space then no carry is generated.
Clear[Base2CarryQ]
Base2CarryQ[a_Integer, b_Integer, reg_Integer : 8] := Module[
{abin = IntegerDigits[a, 2, reg]
, bbin = IntegerDigits[b, 2, reg]
},
Echo[abin];
Echo[bbin];
FromDigits[Echo@IntegerDigits[a b, 2, reg], 2] != a b
]
Usage: (remove Echo statements if not needed)
a = 2;
b = 128;
Base2CarryQ[a, b]
(* carry out from the
default 8 bit register *)
Base2CarryQ[a, b, 9]
(* carry out? of a
9-bit register *)
