Recursive functions, successor function

Question

How to show that the power function $\displaystyle A=2^{m^2}$ is primitive recursive based on successor function?

Thanks much in advance!!!

Answer 1

the squaring function $n \mapsto n^2$ is primitive recursive because you can define addition by primitive recursion, and multiplication in terms of addition by primitive recursion.

the exponentiation function $o \mapsto 2^o$ is primitive recursive defined by primitive recursion in terms of multiplication.

the composite of these is your function, and composite of primitive recursive is primitive recursive.