## URM program to compute n^m (n to the power of m)

For n≠0 and m≠0,

1 S(6)

2 S(7)

3 C(1,3)

4 C(1,4)

5 J(2,7,19)

6 J(4,6,14)

7 J(3,5,11)

8 S(1)

9 S(5)

10 J(1,1,7)

11 S(6)

12 Z(5)

13 J(1,1,6)

14 C(1,3)

15 S(7)

16 Z(6)

17 S(6)

18 J(1,1,5)

n | m | 0 | 0 | 0 | 0 | 0 |

n | m | n | n | 0 | 1 | 1 |

n+1 | m | n | n | 1 | 1 | 1 |

n+2 | m | n | n | 2 | 1 | 1 |

n+3 | m | n | n | 3 | 1 | 1 |

… | … | … | … | … | … | … |

n+n=2n | m | n | n | n | 1 | 1 |

2n | m | n | n | 0 | 2 | 1 |

2n+1 | m | n | n | 1 | 2 | 1 |

2n+2 | m | n | n | 2 | 2 | 1 |

… | … | … | … | … | … | … |

2n+n=3n | m | n | n | n | 2 | 1 |

3n | m | n | n | 0 | 3 | 1 |

… | … | … | … | … | … | … |

… | … | … | … | … | … | … |

n(n)=n^{2} |
m | n | n | n | n-1 | 1 |

n^{2} |
m | n | n | 0 | n | 1 |

n^{2} |
m | n^{2} |
n | 0 | 1 | 2 |

n^{2}+1 |
m | n^{2} |
n | 1 | 1 | 2 |

n^{2}+2 |
m | n^{2} |
n | 2 | 1 | 2 |

… | … | … | … | … | … | … |

n^{2}+n^{2}=2n^{2} |
m | n^{2} |
n | n^{2} |
1 | 2 |

2n^{2} |
m | n^{2} |
n | 0 | 2 | 2 |

… | … | … | … | … | … | … |

… | … | … | … | … | … | … |

n(n^{2})=n^{3} |
m | n^{2} |
n | n^{2} |
n | 2 |

n^{3} |
m | n^{2} |
n | 0 | n | 2 |

n^{3} |
m | n^{3} |
n | 0 | 1 | 3 |

… | … | … | … | … | … | … |

… | … | … | … | … | … | … |

… | … | … | … | … | … | … |

n(n^{m-1})=n^{m} |
m | n^{m-1} |
n | n^{m-1} |
n | m-1 |

n^{m} |
m | n^{m} |
n | 0 | 1 | m |

Example 1.

2 | 4 | 0 | 0 | 0 | 0 | 0 |

2 | 4 | 2 | 2 | 0 | 1 | 1 |

3 | 4 | 2 | 2 | 1 | 1 | 1 |

4 | 4 | 2 | 2 | 2 | 1 | 1 |

4 | 4 | 2 | 2 | 0 | 2 | 1 |

4 | 4 | 4 | 2 | 0 | 1 | 2 |

5 | 4 | 4 | 2 | 1 | 1 | 2 |

6 | 4 | 4 | 2 | 2 | 1 | 2 |

7 | 4 | 4 | 2 | 3 | 1 | 2 |

8 | 4 | 4 | 2 | 4 | 1 | 2 |

8 | 4 | 4 | 2 | 0 | 2 | 2 |

8 | 4 | 8 | 2 | 0 | 1 | 3 |

9 | 4 | 8 | 2 | 1 | 1 | 3 |

10 | 4 | 8 | 2 | 2 | 1 | 3 |

11 | 4 | 8 | 2 | 3 | 1 | 3 |

12 | 4 | 8 | 2 | 4 | 1 | 3 |

13 | 4 | 8 | 2 | 5 | 1 | 3 |

14 | 4 | 8 | 2 | 6 | 1 | 3 |

15 | 4 | 8 | 2 | 7 | 1 | 3 |

16 | 4 | 8 | 2 | 8 | 1 | 3 |

16 | 4 | 8 | 2 | 0 | 2 | 3 |

16 | 4 | 16 | 2 | 0 | 1 | 4 |

Example 2.

3 | 3 | 0 | 0 | 0 | 0 | 0 |

3 | 3 | 3 | 3 | 0 | 1 | 1 |

4 | 3 | 3 | 3 | 1 | 1 | 1 |

5 | 3 | 3 | 3 | 2 | 1 | 1 |

6 | 3 | 3 | 3 | 3 | 1 | 1 |

6 | 3 | 3 | 3 | 3 | 2 | 1 |

6 | 3 | 3 | 3 | 0 | 2 | 1 |

7 | 3 | 3 | 3 | 1 | 2 | 1 |

8 | 3 | 3 | 3 | 2 | 2 | 1 |

9 | 3 | 3 | 3 | 3 | 2 | 1 |

9 | 3 | 3 | 3 | 3 | 2 | 1 |

9 | 3 | 3 | 3 | 0 | 3 | 1 |

9 | 3 | 9 | 3 | 0 | 1 | 2 |

10 | 3 | 9 | 3 | 1 | 1 | 2 |

11 | 3 | 9 | 3 | 2 | 1 | 2 |

12 | 3 | 9 | 3 | 3 | 1 | 2 |

13 | 3 | 9 | 3 | 4 | 1 | 2 |

14 | 3 | 9 | 3 | 5 | 1 | 2 |

15 | 3 | 9 | 3 | 6 | 1 | 2 |

16 | 3 | 9 | 3 | 7 | 1 | 2 |

17 | 3 | 9 | 3 | 8 | 1 | 2 |

18 | 3 | 9 | 3 | 0 | 2 | 2 |

19 | 3 | 9 | 3 | 1 | 2 | 2 |

20 | 3 | 9 | 3 | 2 | 2 | 2 |

21 | 3 | 9 | 3 | 3 | 2 | 2 |

22 | 3 | 9 | 3 | 4 | 2 | 2 |

23 | 3 | 9 | 3 | 5 | 2 | 2 |

24 | 3 | 9 | 3 | 6 | 2 | 2 |

25 | 3 | 9 | 3 | 7 | 2 | 2 |

26 | 3 | 9 | 3 | 8 | 2 | 2 |

27 | 3 | 9 | 3 | 9 | 2 | 2 |

27 | 3 | 9 | 3 | 0 | 3 | 2 |

27 | 3 | 27 | 3 | 0 | 1 | 3 |

For any positive integers n and m,

1 J(1,3,3)

2 J(2,3,23)

3 J(2,3,22)

4 S(6)

5 S(7)

6 C(1,3)

7 C(1,4)

8 J(2,7,25)

9 J(4,6,17)

10 J(3,5,14)

11 S(1)

12 S(5)

13 J(1,1,10)

14 S(6)

15 Z(5)

16 J(1,1,9)

17 C(1,3)

18 S(7)

19 Z(6)

20 S(6)

21 J(1,1,8)

22 J(1,1,22)

23 Z(1)

24 S(1)

Note: if n=m=0, the answer is undetermined and the program never stops.