Exponentiation ($a^b$)

For a nonnegative integer exponent, exponentiation iterates multiplication: $a^0=1$ and $a^{b+1}=a{a}^b$. This convention defines $0^0=1$ in primitive recursive arithmetic. Extensions to other exponents may use different domains and conventions.

A value of the form $a^5$, the product of five copies of $a$.

A value of the form $a^4$, the product of four copies of $a$.

Repeated exponentiation with right association: ${}^{1}a=a$ and ${}^{h+1}a=a^{{}^{h}a}$. For example, ${}^{3}a=a^{a^a}$, rather than $(a^a{)}^a$. Computing its remainder modulo an integer often uses a chain of smaller exponent moduli; applying Euler's theorem requires checking its coprimality hypothesis at every level.

An addition chain for a positive integer $k$ is $1=a_0<a_1<\cdots <a_r=k$, where each $a_t$ is the sum of two earlier entries, which may coincide. It computes $x^k$ in $r$ multiplications by reusing previously computed powers. For example, $1,2,3,6,12,15$ computes $x^{15}$ in five multiplications.

An addition chain restricted to updates $a_{t+1}=a_t+a_i$ with $0\le i\le t$. Each new power uses the most recently computed power as one factor. This restriction reduces the search space, but a shortest star chain need not be a shortest unrestricted addition chain for every target.

The cube of a number $x$ is $x^3=x\cdot x\cdot x$, using the exponent three. For example, $2^3=8$ and $(-2{)}^3=-8$.

A value $n^3$ for an integer $n$.

An exponent specifies the power in exponentiation: in $a^b$, $a$ is the base and $b$ is the exponent. For a positive integer exponent, $2^3=2\cdot 2\cdot 2=8$. For a positive real number base, a real exponent is defined by $a^b=\exp (b\log a)$ using the exponential function and natural logarithm. The Lp norm uses its exponent $p$ to determine how strongly large absolute values affect the result.

The square of a number $a$ is the product $a\cdot a$. For a real number, it is nonnegative. The sum of squares of real coordinates is the squared Euclidean norm of the corresponding vector.

The hyperoperation hierarchy successively iterates basic arithmetic operations. In the usual indexing, successor is followed by addition, multiplication, exponentiation and their higher iterates. A sequence beginning with addition shifts this index by one. Each fixed rank is a primitive recursive function; this does not imply that the evaluator taking the rank as an additional variable is primitive recursive.