Repeat    If x mod m = 0   x=x/m  else x = x*p + m - x mod m until x = 1

This problem uses the same logic as the  3x+1 problem. in that when a chosen divisor does not divide x then an appropriate calculation is made to make x divisible by the divisor. The algorithm tends to converge for any x when m is prime and p = m+1. In this case x = km+r for some k and remainder r. We proceed with the algorithm  and note

  (1)     x = (km+r)(m+1) + m - r .

Multiplying this out we get

       x = (km^2+km +rm + r + m - r  = hm for some h. Thus the operation (1)  transforms x into a multiple of m.  

We then divide x by m and repeat the process. Eventually we get to the value of 1. For example the iteration for x= 2,m=17,p=18 is

51 3 68 4 85 5 102 6 119 7 136 8 153 9 170 10 187 11 204 12 221 13 238 14 255 15  272 16 289 17 1  

See Documents for xdivmxp-r.exe

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