No. Integer arithmetic of any size can be implemented no matter what word size the processor uses. This is done by using both the integer arithmetic instructions of the processor (if any) and bit manipulation instructions.

For example: 16-bit microprocessors can do 64-bit integer arithmetic. This is done by using several 16-bit machine operations for each 64-bit operation and putting the results together with bit-wise logic operations.

As a more modern example: Pentium-1 and Pentium-4 processors can run the same programs. One (the P-4) has a faster machine cycle than the other. And one (the P-4) has more types of machine instructions than the other. If you have a C program that computes something, both processors can run it, and whatever it computes comes out the same on both (assuming appropriate compilers). The running time would be far longer on the P-1 than on the P-4, but running time is not part of the definition.

Sometimes the result a program produces
depends on the compiler.
For example, different compilers for C use
different numbers of bits for
the data type `int`

.
But that is an effect of the compiler,
not of the processor.
All that matters for processor "power" is that
it is possible to translate
identical programs into machine
language appropriate for each processor and that
these machine language programs produce the same result
on each processor.

In 1952 the SWAC digital computer,
once the fastest in the world,
was programmed to
find perfect numbers.
A perfect number is an integer whose integer divisors sum up to the number.
For example, `6`

is perfect because `6`

is divided by
`1, 2,`

and `3`

and `6 = 1 + 2 + 3`

.
Other perfect numbers are 28, 496, and 8128.

After much computation SWAC found the perfect number 33550336. Wiki article on SWAC.

Will a modern Pentium processor find the same perfect numbers as the SWAC?